Local Well-Posedness for the Motion of a Compressible, Self-Gravitating Liquid with Free Surface Boundary

Abstract

We establish the local well-posedness for the free boundary problem for the compressible Euler equations describing the motion of liquid under the influence of Newtonian self-gravity. We do this by solving a tangentially-smoothed version of Euler’s equations in Lagrangian coordinates which satisfies uniform energy estimates as the smoothing parameter goes to zero. The main technical tools are delicate energy estimates and optimal elliptic estimates in terms of boundary regularity, for the Dirichlet problem and Green’s function.

Appendices

Appendix A: Fractional Tangential Derivatives and Tangental Smoothing

There is a family of open sets $U_\mu$, $\mu \!=\!1,\dots ,N$ that cover $\partial \Omega$ and onto diffeomorphisms $\Phi _\mu \!:\! (-1,\!1)^2 \!\rightarrow \!U_\mu$. We fix a collection of cutoff functions $\chi _{\mu }\!{}_{\!}:\!\partial \Omega \! \rightarrow \!\mathbb {R}$ so that $\chi _\mu ^2$ form a partition of unity and ${\text {supp}} \chi _\mu \!{}_{\!}\subset \! U_\mu$, as well as two other families of cutoff functions such that $\widetilde{\chi }_\mu \! \equiv \! 1$ on ${\text {supp}} \chi _\mu$, $\overline{\chi }_\mu \!\equiv \! 1$ on ${\text {supp}} \widetilde{\chi }_\mu$ and ${\text {supp}}\overline{\chi }_\mu \!\!\subset \!U_\mu$. Recalling that $\Omega$ is the unit ball, we set $W_{\!\mu } \!=\! \{r\omega , r \!\in \! (1/2, 1], \omega \!\in \! U_{\!\mu }\}$ for $\mu \!= \!1,{}_{\!}...,{}_{\!} N$ and let $W_0$ be the ball of radius $3/{}_{\!}4$ so that $\{W_{\!\mu }\}_{\mu = 0}^N$ covers $\Omega$. Writing $\Psi _\mu (z,z_3) = z_3 \Phi _\mu (z)$, $\Psi _\mu$ is a diffeomorphism from $(-1,{}_{\!}1)^2\! \times (1{}_{\!}/2,{}_{\!}1]$ to $W_{\!\mu }$. Let $\zeta \!:\![0,\!1] \rightarrow \mathbb {R}$ be a bump function so that $\zeta (r) \!= \!1$ when $1/2\! \le \! r\! \le \!1$ and $\zeta (r)\! = \!0$ when $r \!<\! 1/4$. We extend the above cutoffs to $\Omega$ by setting $\chi _\mu (y) = \chi _\mu (y/|y|) \zeta (|y|)$ for $\mu = 1,..., N$ and $\chi _0 =\! 1\!-\!\zeta$, and we similarly extend $\widetilde{\chi }_\mu$ and $\overline{\chi }_\mu$. We abuse notation by writing $\chi _\mu$ also for the function $\chi _\mu \circ \Psi _\mu$

1.1 A.1: Fractional Derivatives

For a function $F: \mathbb {R}^2 \rightarrow \mathbb {R}$, we set

$$\begin{aligned} \langle \partial _{\theta {}_{\!}} \rangle ^{\!1{}_{\!}/2}F(z) = \int _{\mathbb {R}^2} \langle \xi \rangle ^{1/2} \hat{F}(\xi ) e^{iz\cdot \xi }\,\mathrm{d}\xi , \quad \text {where } \hat{F}(\xi ) = \int _{\mathbb {R}^2} e^{-iz\cdot \xi } F(z)\, \mathrm{d}z. \end{aligned}$$

Given a function $f: \Omega \rightarrow \mathbb {R}$, we define $\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu f: \Omega \rightarrow \mathbb {R}$ for $\mu = 1,..., N$ by

$$\begin{aligned} \langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu f = \widetilde{\chi }_\mu (\langle \partial _{\theta {}_{\!}} \rangle ^{\!1{}_{\!}/2}f_\mu )\circ \Psi _\mu ^{-1},\quad \text {where}\quad f_\mu = (\chi _\mu f)\circ \Psi _\mu :\mathbb {R}^2 \rightarrow \mathbb {R}. \end{aligned}$$

(A.1)

With the cutoff function $\zeta$ defined above, we let ${\mathcal {T}^{}}$ denote the following family of vector fields, which span the tangent space to the boundary and in the interior span the full tangent space

$$\begin{aligned} \zeta (y) (y^a\partial _{y^b} - y^b \partial _{y^a}), \quad (1 - \zeta (y)) \partial _{y^a}, \quad a,b =1,2,3. \end{aligned}$$

We work in terms of the following Sobolev norms, for $s \in \mathbb {R}$:

$$\begin{aligned} ||f||_{H^s(\partial \Omega )}^2 = {\sum }_{\mu = 1}^N ||\langle \partial _{\theta {}_{\!}} \rangle ^{s} f_\mu ||_{L^2(\mathbb {R}^2)}^2 = {\sum }_{\mu = 1}^N \int _{\mathbb {R}^2} |\langle \xi \rangle ^{s} \hat{f}_\mu (\xi )|^2\, \mathrm{d}\xi , \end{aligned}$$

(A.2)

and if $s \in \mathbb {R}, k \in \mathbb {N}$, we set

$$\begin{aligned} ||f||_{H^{(k,s)}(\Omega )}^2 = {\sum }_{|I| \le k} \int _0^1 ||\partial _y^I (\zeta f)(r,\cdot )||_{H^s(\partial \Omega )}^2 \, r^2 dr + ||(1-\zeta ) f||_{H^{k+s}(\Omega )}^2, \end{aligned}$$

where for non-integer s, $H^{k+s}(\Omega )$ is defined in the usual way by taking the Fourier transform in all variables. We collect here the basic properties of the operators $\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu$ and the norms $H^s(\partial \Omega ), H^{(k,s)}(\Omega )$.

Lemma A.1

If $T \in {\mathcal {T}^{}}$, then

$$\begin{aligned}&\Big | \int _{\partial \Omega } f T g\, \mathrm{d}S(y)\Big | \le C||f||_{H^{1/2}(\partial \Omega )} ||g||_{H^{1/2}(\partial \Omega )},\nonumber \\&\qquad \Big | \int _{\Omega } f T g\, \mathrm{d}y \Big | \le C||f||_{H^{(0,1/2)}(\Omega )} ||g||_{H^{(0,1/2)}(\Omega )}. \end{aligned}$$

(A.3)

In addition, with $\Sigma = \partial \Omega$ or $\Omega$,

$$\begin{aligned} ||\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu (fg) - f \langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu g||_{L^2(\Sigma )} \le C ||f||_{H^2(\Sigma )} ||g||_{L^2(\Sigma )}, \end{aligned}$$

(A.4)

and, with notation as in (3.5) and $T^I \in \mathcal {D}^k$ or ${\mathcal {T}^{}}^k$,

$$\begin{aligned} ||\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu (T^I f) - T^I \langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu f||_{L^2(\Sigma )} \le C||f||_{H^{k}(\Sigma )}. \end{aligned}$$

(A.5)

In particular, if $||\widetilde{x}||_{H^3(\Omega )} \le M$ then

$$\begin{aligned} ||\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu \widetilde{\partial }f - \widetilde{\partial }\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu f||_{L^2(\Sigma )} \le C (M) ||f||_{H^1(\Sigma )}. \end{aligned}$$

(A.6)

These estimates all rely on the following “Leibniz rule”. This lemma and its proof can be found in [21].

Lemma A.2

If $F, G: \mathbb {R}^2 \rightarrow \mathbb {R}$ have compact support, then

$$\begin{aligned} ||\langle \partial _{\theta {}_{\!}} \rangle ^{\!1{}_{\!}/2}(FG) - F \langle \partial _{\theta {}_{\!}} \rangle ^{\!1{}_{\!}/2}G||_{L^2(\mathbb {R}^2)} \le C ||F||_{H^2(\mathbb {R}^2)} ||G||_{L^2(\mathbb {R}^2)}. \end{aligned}$$

(A.7)

Proof

By the elementary estimate $|\langle \xi \rangle ^{1/2} - \langle \xi -\eta \rangle ^{1/2}| \le C \langle \eta \rangle ^{1/2}$, we have

$$\begin{aligned}&|\langle \xi \rangle ^{1/2} \widehat{ FG} (\xi ) - \widehat{ (F \langle \partial _{\theta {}_{\!}} \rangle ^{\!1{}_{\!}/2}G)}(\xi )|^2 \lesssim \Big (\int _{\mathbb {R}^2} \!\!\!\langle \eta \rangle ^{1/2} |\hat{F}(\eta )| |\hat{G}(\xi \!-\!\eta )|\, d\eta \Big )^2\!\! \\&\quad \lesssim \int _{\mathbb {R}^2} \!\!\!\langle \eta \rangle ^{4} |\hat{F}(\eta )|^2\, d\eta \int _{\mathbb {R}^2} \!\!\! \langle \eta \rangle ^{-3} |\hat{G}(\xi \!-\!\eta )|^2\, d\eta . \end{aligned}$$

Integrating in $\xi$, changing variables, and using the fact that $\int _{\mathbb {R}^2} \langle \xi -\eta \rangle ^{-3} \, \mathrm{d}\xi \! \le \!C$, we have

$$\begin{aligned}&||\langle \xi \rangle \widehat{FG} - \widehat{(F \langle \partial _{\theta {}_{\!}} \rangle ^{\!1{}_{\!}/2}G)}||_{L^2(\mathbb {R}^2)}^2 \le C ||F||_{H^2(R)}^2 \int _{\mathbb {R}^2} \int _{\mathbb {R}^2} \langle \xi \\&\quad -\eta \rangle ^{-3} |\hat{G}(\eta )|^2\, d\eta \, \mathrm{d}\xi \le C||F||_{H^2(R)} ||G||_{L^2(R)}. \end{aligned}$$

The result now follows from Plancherel’s theorem. $\quad \square$

Proof of Lemma A.1

Since $\sum \chi _\mu ^2 = 1$, we have

$$\begin{aligned}&\int _{\partial \Omega } fT g \, \mathrm{d}S(y) = {\sum }_{\mu = 1}^N \int _{\partial \Omega } (\chi _\mu f) (\chi _\mu T g)\, \mathrm{d}S(y)\\&\quad = {\sum }_{\mu = 1}^N \int _{\partial \Omega } \chi _\mu f T(\chi _\mu g) \, \mathrm{d}S(y) - \int _{\partial \Omega } \chi _\mu f g T\chi _\mu \, \mathrm{d}S(y). \end{aligned}$$

The second term is bounded by $C||f||_{L^2(\partial \Omega )} ||g||_{L^2(\partial \Omega )}$. To deal with the first term, we use (A.1) and write

$$\begin{aligned} \int _{\partial \Omega } \chi _\mu f T(\chi _\mu g) \, \mathrm{d}S(y) = \int _{\mathbb {R}^2} f_\mu T^\alpha \partial _{z^\alpha } g_\mu |\det {\Phi '_\mu }| \,\mathrm{d}z, \quad \text {where}\quad T= T^\alpha \partial _{z^\alpha }. \end{aligned}$$

With $F^\alpha = f_\mu T^\alpha |\det {\Phi '_\mu }|$ and $G = g_\mu$, by Plancherel’s theorem we have

$$\begin{aligned} \int _{\mathbb {R}^2} F^\alpha (z) \partial _{z^\alpha } G(z) \mathrm{d}z = \int _{\mathbb {R}^2} \hat{F}^\alpha (\xi ) i\xi _\alpha \hat{G}(\xi )\, \mathrm{d}\xi \le || \langle \xi \rangle ^{1/2} \hat{F}||_{L^2(\mathbb {R}^2)} || \langle \xi \rangle ^{1/2} \hat{G}||_{L^2(\mathbb {R}^2)}. \end{aligned}$$

By (A.7) and (A.2), this is bounded by $(||\langle \partial _{\theta {}_{\!}} \rangle ^{\!1{}_{\!}/2}\!f_{\mu \!}||_{L^2(R)}\! +\! ||f||_{L^2(\partial \Omega )})||g||_{{}_{\!}H^{1\!/2}(\partial \Omega )}$. The case $\Sigma \!= \! \Omega$ is similar.

We now prove (A.4). Writing $\overline{f}_\mu =\overline{\chi }_\mu f\circ \Psi _\mu$, where $\overline{\chi }_\mu \equiv 1$ in the support of $\widetilde{\chi }_\mu$ in (A.1), we have

$$\begin{aligned}&||{}_{\!}\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu (f{}_{\!}g_{{}_{\!}})\! -\! f \langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu {}_{\!}g||_{L^2{}_{\!}(\partial \Omega )} \! \lesssim \! ||{}_{\!}\langle \partial _{\theta {}_{\!}} \rangle ^{\!1{}_{\!}/2}({}_{\!}\overline{f}_{\!\!\mu } g_{\mu \!})\! - \!\overline{f}_{\!\!\mu \!} \langle \partial _{\theta {}_{\!}} \rangle ^{\!1{}_{\!}/2}{}_{\!}g_{\mu \!}||_{L^2{}_{\!}(\mathbb {R}^{2\!})} \!\\&\quad \lesssim \! \Vert {}_{\!}\overline{f}_{\!\!\mu \!}\Vert _{H^2(\mathbb {R}^{2\!})} \Vert g_{\mu \!}\Vert _{L^2(\mathbb {R}^{2\!})} \!\lesssim \! ||f||_{H^2(\!\partial \Omega )}||g_{{}_{\!}}||_{L^2(\!\partial \Omega )}, \end{aligned}$$

by (A.7), which gives (A.4) for $\Sigma \!= \!\partial \Omega$. The case $\Sigma \! =\! \Omega$ follows from the case $\Sigma \!=\! \partial \Omega$ by the definition (A.2).

We now prove (A.5). We first prove the case $k \!=\! 1$ with $T\in {\mathcal {T}^{}}$ and $\Sigma = \partial \Omega$. Since $\partial _{z^\alpha } \langle \partial _{\theta {}_{\!}} \rangle ^{\!1{}_{\!}/2}= \langle \partial _{\theta {}_{\!}} \rangle ^{\!1{}_{\!}/2}\partial _{z^\alpha }$:

$$\begin{aligned}&T^\alpha \partial _{z^\alpha } \langle \partial _{\theta {}_{\!}} \rangle ^{\!1{}_{\!}/2}f_\mu - \langle \partial _{\theta {}_{\!}} \rangle ^{\!1{}_{\!}/2}(T f)_\mu = T^\alpha \langle \partial _{\theta {}_{\!}} \rangle ^{\!1{}_{\!}/2}(\partial _\alpha f_\mu ) \nonumber \\&\quad - \langle \partial _{\theta {}_{\!}} \rangle ^{\!1{}_{\!}/2}(T^\alpha \partial _\alpha f_\mu )+\langle \partial _{\theta {}_{\!}} \rangle ^{\!1{}_{\!}/2}((T^\alpha \partial _\alpha \chi _\mu )f), \end{aligned}$$

(A.8)

Applying (A.7), the $L^2$ norm of the right-hand side is bounded by $C ||f||_{H^1(\partial \Omega )}$. The commutator of $T \langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu \! f\! -\! \langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu T\! f$ just contribute an additional term $(T\widetilde{\chi }_\mu )\langle \partial _{\theta {}_{\!}} \rangle ^{\!1{}_{\!}/2}\!f_\mu$ compared to (A.8) and (A.5) follows.

To prove (A.5) when $T = \partial _{y^a}$ for some $a = 1,2,3$ and $\Sigma = \partial \Omega$, close to the boundary we write $\partial _{y^a} = \sum _{T \in {\mathcal {T}^{}}} c_a^T(y) T\! + \!c(y) \partial _r$ for some smooth functions $c_a^T$ and c. By what we have just proven and (A.4), it is enough to prove the estimate with T replaced by $\partial _r$. This follows from the definition after noting that close to the boundary, the cutoff functions $\widetilde{\chi }_\mu , \chi _\mu$ are independent of r. The case $|I| \ge 2$ follows similarly. $\quad \square$

The operators $\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu$ can be used to control fractional Sobolev norms as follows:

Lemma A.3

We have

$$\begin{aligned} ||(1-\widetilde{\chi }_\mu ) \langle \partial _{\theta {}_{\!}} \rangle ^{\!1{}_{\!}/2}f_\mu ||_{L^2(\mathbb {R}^2)}\lesssim ||f_\mu ||_{L^2(\mathbb {R}^2)}. \end{aligned}$$

Moreover, there are constants $0<C_1< C_2<\infty$ so that

$$\begin{aligned}&C_1 \bigg ({\sum }_{\mu = 1}^N ||\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu f||_{L^2(\partial \Omega )}+||f||_{L^2(\partial \Omega )}\bigg ) \le ||f||_{H^{1/2}(\partial \Omega )} \\&\quad \le C_2 \bigg ( {\sum }_{\mu = 1}^N ||\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu f||_{L^2(\partial \Omega )} +||f||_{L^2(\partial \Omega )}\bigg ). \end{aligned}$$

The same estimate holds with $\partial \Omega$ replaced by $\Omega$ and $H^{1/2}(\partial \Omega )$ replaced with $H^{(0,1/2)}(\Omega )$.

Proof

Since $\widetilde{\chi }_\mu =1$ on the support of $\chi _\mu$ and hence on the support of $f_\mu$ it follows from (A.7) that

$$\begin{aligned}&||(1-\widetilde{\chi }_\mu ) \langle \partial _{\theta {}_{\!}} \rangle ^{\!1{}_{\!}/2}f_\mu ||_{L^2(R)} =|| \langle \partial _{\theta {}_{\!}} \rangle ^{\!1{}_{\!}/2}(\widetilde{\chi }_\mu f_\mu )-\widetilde{\chi }_\mu \langle \partial _{\theta {}_{\!}} \rangle ^{\!1{}_{\!}/2}f_\mu ||_{L^2(R)} \\&\quad \le C ||f_\mu ||_{L^2(R)}\le C ||f||_{L^2(\partial \Omega )}. \end{aligned}$$

$\square$

1.2 A.2: Tangential Smoothing

Let $\varphi \!:\! \mathbb {R}^2 \!\!\rightarrow \! \mathbb {R}$ be an even smooth function, supported in $R = (-1,1)^2$, with $\int _{\mathbb {R}^2} \!\varphi =\! 1$ and define the smoothing operator

$$\begin{aligned} T_\varepsilon f(z) = \int _{\mathbb {R}^2} \varphi _{\varepsilon }(z-z') f(z') \mathrm{d}z',\qquad \text {where}\qquad \varphi _{\varepsilon }(z)\! =\! {\varepsilon ^{-2}} \varphi \big ({z}\!/{\varepsilon }\big ). \end{aligned}$$

Because $\varphi$ is even, $T_\varepsilon$ is symmetric; for any functions $f, g: \mathbb {R}^2 \rightarrow \mathbb {R}$ we have

$$\begin{aligned}&\int _{\mathbb {R}^2} T_\varepsilon f(z) g(z) \mathrm{d}z = \int _{\mathbb {R}^2} \int _{R} \varphi _\varepsilon (z-z') f(z') g(z) \mathrm{d}z'\, \mathrm{d}z \\&\quad = \int _{\mathbb {R}^2} f(z) T_\varepsilon g(z) \mathrm{d}z. \end{aligned}$$

From the fact that $\Vert \partial ^k \varphi _\varepsilon \Vert _{L^1}\lesssim \varepsilon ^{-k}$ it follows that, for $k\ge m$,

$$\begin{aligned} \Vert T_\varepsilon f\Vert _{H^k}\lesssim \varepsilon ^{m-k} \Vert f\Vert _{H^m}. \end{aligned}$$

(A.9)

Furthermore, we have

$$\begin{aligned} |T_\varepsilon (fg)(z) - fT_\varepsilon (g)(z)| \le C \varepsilon ||f||_{C^1(R)} ||g||_{L^2(R)}, \end{aligned}$$

(A.10)

which follows from the fact that $|z'| \le \varepsilon$ in the support of $\varphi _\varepsilon$, after writing

$$\begin{aligned} T_\varepsilon (fg)(z) - f(z)T_\varepsilon (g)(z) = \int _{\mathbb {R}^2} \varphi _\varepsilon (z')g(z-z')\big ( f(z-z') - f(z)\big ) \mathrm{d}z'. \end{aligned}$$

(A.11)

Moreover from using (A.20) and Minkowski’s integral inequality in (A.11) with $g\!=\!1$ it follows that

$$\begin{aligned} \Vert T_\varepsilon f-f\Vert _{H^{k}}\lesssim \varepsilon \Vert f\Vert _{H^{k+1}}. \end{aligned}$$

(A.12)

For a linear operator $T'$ defined in coordinate charts we define a global operator T by

$$\begin{aligned}&T\!f\!=\!\sum T_{\!\mu } f,\quad \text {where}\quad T_{\!\mu } f \!= \chi _\mu \big (m_\mu ^{-1}T'\big [m_\mu f_{\!\mu } \big ]\big )\circ \Psi _{\!\mu }^{-1}\!\!,\qquad f_{\!\mu } \nonumber \\&\quad \!=\! (\chi _\mu f)\circ \Psi _{\!\mu },\quad m_\mu \!=\!r|\!\det {\Phi _{\!\mu }^\prime }|^{1/2}\!\!. \end{aligned}$$

(A.13)

Then T is symmetric with the measure $\mathrm{d}y$ if $T'$ is with the measure $\mathrm{d}z$ is since $\mathrm{d}S(\omega )=m_\mu ^2 \mathrm{d}z$. With notation as in (A.13), the smoothing operators we consider on $\Omega$ or $\partial \Omega$ are then defined by

$$\begin{aligned} J_\varepsilon f = {\sum }_{\mu = 0}^N T_{\varepsilon ,\mu } f, \qquad S_\varepsilon f = J_\varepsilon J_\varepsilon f={\sum }_{\mu ,\nu = 0}^N T_{\varepsilon ,\nu } T_{\varepsilon ,\mu } f. \end{aligned}$$

Since $T_\varepsilon$ is symmetric $J_\varepsilon$ is as well, w.r.t. $\mathrm{d}y$.

The smoothing operator has the following important properties:

Lemma A.4

If $f, g : \Omega \rightarrow \mathbb {R}$, then with $\Sigma = \partial \Omega$ or $\Omega$

$$\begin{aligned}&||J_\varepsilon (fg) - f J_\varepsilon (g)||_{L^2(\Sigma )} \le C \varepsilon ||f||_{C^1(\Sigma )} ||g||_{L^2(\Sigma )}. \end{aligned}$$

(A.14)

$$\begin{aligned}&\Big | \int _{\partial \Omega }\big ( f (S_\varepsilon g) - (J_\varepsilon f)(J_\varepsilon g) \big )\,\widetilde{\nu } \mathrm{d}S(y)\Big | \le C \varepsilon ||\widetilde{\nu }||_{C^1(\partial \Omega )} ||f||_{L^2(\Omega )} ||g||_{L^2(\partial \Omega )}, \nonumber \\ \end{aligned}$$

(A.15)

$$\begin{aligned}&\Big | \int _{\Omega }\big ( f (S_\varepsilon g) - (J_\varepsilon f)(J_\varepsilon g) \big )\,\widetilde{\kappa }\mathrm{d}y\Big | \le C \varepsilon ||\widetilde{\kappa }||_{C^1(\Omega )} ||f||_{L^2(\Omega )} ||g||_{L^2(\Omega )}. \end{aligned}$$

(A.16)

Further, if $T^I \in {\mathcal {T}^{}}^k$ for $k \ge 0$,

$$\begin{aligned}&||T^I J_\varepsilon f - J_\varepsilon \big (T^I f\big )||_{L^2(\Sigma )} \le C ||f||_{H^{k-1}(\Sigma )}, \end{aligned}$$

(A.17)

$$\begin{aligned}&||\langle \partial _{\theta {}_{\!}} \rangle ^{1/2}_\mu J_\varepsilon f - J_\varepsilon \big (\langle \partial _{\theta {}_{\!}} \rangle ^{1/2}_\mu f\big )||_{L^2(\Sigma )} \le C ||f||_{L^2(\Sigma )}. \end{aligned}$$

(A.18)

Proof

The estimate (A.14) is a straightforward consequence of (A.10).

To prove (A.15) and (A.16) note first that $J_\varepsilon$ is symmetric w.r.t. to the measure $\mathrm{d}S(y)$, since $m_\mu ^2 \mathrm{d}S(y)=\mathrm{d}z$:

$$\begin{aligned}&\int _{\partial \Omega } \!\!\!\! f {}_{\!} J_{\varepsilon } g\, \mathrm{d}S \!= \!\!{\sum }_{\mu \!}\!\int _{\partial \Omega } \!\!\!\! f{}_{\!}\chi _{{}_{\!}\mu {}_{\!}} \big ( {}_{\!} m_\mu ^{-\!1} T_{\!\varepsilon } [m_\mu g_{\mu {}_{\!}}]\big ){}_{\!}\circ {}_{\!} \Psi _\mu ^{-\!1\!} \mathrm{d}S \!=\!\! {\sum }_{\mu \!} \!\int _{\!R} \!\! m_\mu f_{\!\mu } T_{\!\varepsilon }[ m_\mu g_{\mu {}_{\!}}] \mathrm{d}z \\&\quad \!=\!\!{\sum }_{\mu \!}\!\int _{\!R} \! T_{\!\varepsilon }[ m_\mu f_{\!\mu {}_{\!}}] m_\mu g_\mu \mathrm{d}z \!=\!\!\int _{\partial \Omega } \!\!\!{}_{\!}J_{\varepsilon {}_{\!}}f\, g\, \mathrm{d}S. \end{aligned}$$

(A.15) follows from this applied to $\widetilde{\nu } f\!$ in place of $f\!$ and then (A.14) with $\widetilde{\nu }$ in place of $f\!$ and $f\!$ in place of g.

Changing coordinates, using that $\partial _z (\varphi _\varepsilon \!* \!F)\! =\! \varphi _\varepsilon \! *\! (\partial _z F)$ for any function $F\!\!:\! (-1,1)^2 \!\rightarrow \mathbb {R}$ and using (A.14), a straightforward calculation as in the proof of Lemma A.1 shows that $||T^I \! J_\varepsilon f - J_\varepsilon T^I \!f||_{L^2(\Sigma )} \lesssim ||f||_{H^{k-1}(\Sigma )}$.

(A.18) follows from that $[\langle \partial _{\theta {}_{\!}} \rangle ^{{}_{\!}1_{\!}/2}\!\!, T_{\!\varepsilon }]\!=\!0$ and $\sum _\nu \chi _\nu ^2\!=\!1$, after repeatedly using (A.4) and (A.14) in

$$\begin{aligned}&\langle \partial _{\theta {}_{\!}} \rangle ^{1/2}_\mu J_\varepsilon f ={\sum }_\nu \widetilde{\chi }_\mu \bigg ( \langle \partial _{\theta {}_{\!}} \rangle ^{1/2}\bigg [ \chi _\mu \chi _\nu m_\nu ^{-1} T_{\!\varepsilon }\big [m_\nu f_{\!\nu } \big ]\bigg ]\bigg )\circ \Psi _{\!\nu }^{-1}, \\&J_\varepsilon \langle \partial _{\theta {}_{\!}} \rangle ^{1/2}_\mu f ={\sum }_\nu \chi _\nu \bigg (m_\nu ^{-1} T_{\!\varepsilon }\bigg [ \chi _\nu m_\nu \widetilde{\chi }_\mu \langle \partial _{\theta {}_{\!}} \rangle ^{1/2}[ f_{\!\mu }] \big ]\bigg ]\bigg )\circ \Psi _{\!\nu }^{-1}. \end{aligned}$$

$\square$

1.3 A.3: Interpolation and Sobolev Inequalities

Here we collect some standard inequalities we will use.

We will use the Sobolev inequalities on both $\Omega$ and $\partial \Omega$. For any tensor field $\alpha$ on either $\Omega \cup \partial \Omega$ or $\partial \Omega$,

$$\begin{aligned} ||\alpha ||_{L^{3p/(3-kp)}(\Omega )}&\le C {\sum }_{|I| \le k}||\partial _y^I \alpha ||_{L^p(\Omega )},&1 \le p< {3}/{k},\nonumber \\ ||\alpha ||_{L^\infty (\Omega )}&\le C {\sum }_{|I| \le k} ||\partial _y^I \alpha ||_{L^p(\Omega )},&k> {3}/{p},\nonumber \\ ||\alpha ||_{L^{2p/(2-kp)}(\partial \Omega )}&\le C {\sum }_{|I| \le k} ||\partial _y^I \alpha ||_{L^p(\partial \Omega )},&1 \le p < {2}/{k},\nonumber \\ ||\alpha ||_{L^\infty (\partial \Omega )}&\le {\sum }_{|I| \le k} ||\partial _y^I \alpha ||_{L^p(\partial \Omega )}.&k> {2}/{p}. \end{aligned}$$

(A.19)

By, e.g. the results in the appendix of [3], the constants above depend only on the injectivity radius of $\Omega$.

We also have the following alternative characterization of the Sobolev spaces

$$\begin{aligned} \Vert D_c^h F\Vert _{L^2}\lesssim \Vert \partial _c F\Vert _{L^2}\lesssim {\sup }_h \Vert D_c^h F\Vert _{L^2} , \quad \text {where}\quad D_c^h F(z) = \big (F(z + h e_c) - F(z)\big )/{h}, \end{aligned}$$

(A.20)

denotes the difference quotient in the direction of a unit vector $e_c$, see [9].

We will also need the trace inequality (see, e.g. [22])

$$\begin{aligned} ||f||_{H^{s-1/2}(\partial \Omega )} \le C ||f||_{H^s(\Omega )},\qquad s>1/2. \end{aligned}$$

(A.21)

We will only apply this when s is a positive integer and in that case the right-hand side is defined in the usual way and the left-hand side is defined by (3.4). We will use the following Sobolev inequalities:

Lemma A.5

If $s \ge 2$, then

$$\begin{aligned} ||f||_{L^\infty (\Omega )} \le C ||f||_{H^s(\Omega )}. \end{aligned}$$

(A.22)

Further, with notation as in Section 3.3, if $s \ge 2$ then

$$\begin{aligned} ||f||_{L^\infty (\Omega )} \le C ||{\mathcal {T}^{}}^{{}_{\,}s} f||_{H^1(\Omega )}. \end{aligned}$$

(A.23)

If $k < {3}/{p}$ and ${1}/{q} = {1}/{p} - {k}/{3}$, then

$$\begin{aligned} ||f||_{L^p(\Omega )} \le C {\sum }_{|I| \le k} || \partial _y^I f||_{L^q(\Omega )}. \end{aligned}$$

(A.24)

Proof

The estimates (A.22) and (A.24) are the usual Sobolev inequalities. The estimate (A.23) follows after applying the one-dimensional Sobolev inequality in the radial direction and the two-dimensional Sobolev inequality in the tangential directions. $\quad \square$

We also have the following product rule:

Lemma A.6

Suppose that $|\partial _y^I D_t^k f| \le K$ in $\Omega$ for all $|I| + k \le 3$. Then, if $k + \ell =s$, we have

$$\begin{aligned} ||fg||_{k,\ell } \le (||f||_{k,\ell } + K) (||g||_{k,\ell } + ||g||_{s-1}). \end{aligned}$$

(A.25)

The right-hand side can also be bounded by $(||f||_s + L) ||g||_s$, but for some our applications it is more useful to keep track of which types of derivatives land on f.

Proof

We need to bound $|| (D_t^{k_1\!} \partial _y^{J_1} \!f) ( D_t^{k_2} \partial _y^{J_2} g)||_{L^2(\Omega )}$ where $k_1\! + \!k_2 \!+\! |J_1| \!+\! |J_2| \!= \! s$. If $k_1\! +\! |J_1| \!\le 3$, we bound this by $||D_t^{k_1\!} \partial _y^{J_1}\! f||_{L^\infty (\Omega )} ||D_t^{k_2} \partial _y^{J_2} g||_{L^2(\Omega )}$ which is bounded by the right-hand side of (A.25). If instead $k_1\! + \! |J_1| \ge 4$, we bound it by $|| D_t^{k_1\!} \partial _y^{J_1}\! f||_{L^2(\Omega )} ||D_t^{k_2\!} \partial _y^{J_2}\! g||_{L^\infty (\Omega )} \le ||f||_{k,\ell } ||g||_{2 + k_2 + |J_2|}$. Since $k_1 + |J_1| \ge 4$ and $k_1 + k_2 + |J_1| + |J_2| = s$, it follows that $2+ k_2 + |J_2| \le s$, as required. $\quad \square$

1.4 A.4: The Extension Operator

Fix an integer $s \ge 0$. Let $\eta = \eta (r)$ be a smooth cutoff function which is one when $r \le 1 + 1/(4+4s)$ and zero when $r \ge 1+ 1/(2+2s)$. Let $\lambda _0,..., \lambda _s$ be the solution to the system ${{\,\mathrm{{\textstyle {\sum }}}\,}}_{j = 0}^s \lambda _j (-(j+1))^\ell = 1$ for $\ell = 0,..., s$. If $f: \Omega \rightarrow \mathbb {R}$, we extend f to a function $Ef = E_sf$ on $\mathbb {R}^3$ by setting $Ef(y) = f(y)$ when $|y| \le 1$ and when $|y| \ge 1$, write $f(y) = f(r, \omega )$ where $r = |y|, \omega = y/|y|\in {\mathbb {S}}^2$ and define

$$\begin{aligned} Ef(r , \omega ) = {\sum }_{j = 0}^s \lambda _j f(r - (j+1)(r-1), \omega ) \eta (r), \quad r \ge 1. \end{aligned}$$

(A.26)

Let $\zeta = \zeta (r)$ be a smooth function with $\zeta (r) = 0, r \le 1/4$ and $\zeta = 1$ for $r \ge 1/2$. For $f:\mathbb {R}^3\rightarrow \mathbb {R}$, we define $||f||_{H^{(k,s)}(\mathbb {R}^3)}^2 = {\sum }_{|I| \le k} \int _0^\infty ||\partial _y^I (\zeta f)(r,\cdot )||_{H^s(\partial \Omega )}^2 \, r^2 dr + ||(1-\zeta )f||_{H^{k+s}(\Omega )}^2,$ and we have

Theorem A.7

Fix $s \ge 2$ and define $E = E_s$ by (A.26). Then E is continuous as a map $H^s(\Omega ) \rightarrow H^s(\mathbb {R}^3)$ and $H^{(s,1/2)}(\Omega ) \rightarrow H^{(s,1/2)}(\mathbb {R}^3)$ and there are constants $0< C_1< C_2 < \infty$ depending only on s so that

$$\begin{aligned} C_1 ||Ef||_{H^{(s,a)} (\mathbb {R}^3)} \le ||f||_{H^{(s,a)}(\Omega )} \le C_2||Ef||_{H^{(s,a)}(\mathbb {R}^3)} \qquad \text { where } a = 0,1/2, \end{aligned}$$

(A.27)

there is a constant C depending only on s so that if T is any vector field on $\mathbb {R}^3$ with $T|_{\Omega } \in {\mathcal {T}^{}}$, then

$$\begin{aligned} ||T Ef||_{H^{(s, a)}(\mathbb {R}^3)} \le C(|| ET f||_{H^{(s,a)}(\mathbb {R}^3)}+|| E f||_{H^{(s, a)}(\mathbb {R}^3)}), \quad \text { where } a=0, 1/2. \end{aligned}$$

(A.28)

Proof

We have

$$\begin{aligned} \partial _r^\ell (Ef)(r,\omega ) = {\sum }_{j = 0}^s \lambda _j \partial _r^\ell f(r - (j+1)(r-1),\omega ) \big (-(j+1)\eta (r)\big )^\ell + g_\ell (r,\omega ),\quad r \ge 1, \end{aligned}$$

where $g_\ell (1,\omega ) \!= \!0$, so by the definition of the $\lambda _j$ and the fact that $\eta (1)\! =\! 1$, it follows that $\partial _r^k(Ef)(1,\omega ) \!= \! \partial _r^k f(1,\omega )$ for $1\! \le \!k \!\le \!s$ and $\omega \! \in \! {\mathbb {S}}^2$. This implies the estimate (A.27). The estimate (A.28) follows from the fact that near the boundary, $T \in {\mathcal {T}^{}}$ commutes with E since $(y^a\partial _b - \partial _b y^a) |y|^2 = 0$.

1.5 A.5: The Green’s Formula

We conclude this section by recording the following Green’s formula which will be frequently used throughout this manuscript. Let $f, g: \mathcal {D}\rightarrow \mathbb {R}$ be $C^1$ functions, then

$$\begin{aligned}&\int _{\Omega } \widetilde{\partial }_i f(\widetilde{x}(t,y)) g(\widetilde{x}(t, y)) \widetilde{\kappa }\mathrm{d}y = \int _{\widetilde{\mathcal {D}}_t} \widetilde{\partial }_i f(\widetilde{x})g(\widetilde{x})d\widetilde{x}\nonumber \\&\quad = -\int _{\widetilde{\mathcal {D}}_t} f(\widetilde{x})\widetilde{\partial }_i g(\widetilde{x})d\widetilde{x}+\int _{\partial \widetilde{\mathcal {D}}_t} N_i f(\widetilde{x})g(\widetilde{x})\mathrm{d}S(\widetilde{x})\nonumber \\&\quad =-\int _{\Omega } f(y)\widetilde{\partial }_i g(y)\widetilde{\kappa }\mathrm{d}y+\int _{\partial \Omega } N_i f(y)g(y)\widetilde{\nu } \mathrm{d}S(y). \end{aligned}$$

(A.29)

Appendix B: Proofs of Elliptic Estimates for the Dirichlet Problem

Here we prove the elliptic estimates we need. We will use these to prove that $\Lambda$ is a continuous map on a certain Banach space and to prove that $\Lambda$ is a contraction, in Section 9. The basic estimates we need for the contraction estimates imply the estimates for the operator norm so we start with the contraction estimates.

Let $V_{{}_{\!}{}_{\!}I}{}_{\!},{\!} V_{{}_{\!}I{\!}I}{\!}:{}_{\!} [0,T] \!\times \!\Omega \!\rightarrow \mathbb {R}^3$ be two vector fields on $\Omega$ and let $\,\widetilde{\!x}_{{}_{\!}I}, \,\widetilde{\!x}_{{}_{\!}I\!I}$ denote their smoothed flows (4.1). Set

$$\begin{aligned} A_{{}_{\!}I\, a}^{\,\,i} = \frac{\partial \,\widetilde{\!x}_{{}_{\!}I}^i}{\partial y^a},\qquad A_{{}_{\!}I\, i}^{\,\,a}= \frac{\partial y^a}{\partial \,\widetilde{\!x}_{{}_{\!}I}^i}\qquad \text {and}\qquad A_{{}_{\!}I{\!}I\, a}^{\,\,\,i}= \frac{\partial \,\widetilde{\!x}_{{}_{\!}I\!I}^i}{\partial y^a},\qquad A_{{}_{\!}I{\!}I\, i}^{\,\,\,a}= \frac{\partial y^a}{\partial \,\widetilde{\!x}_{{}_{\!}I\!I}^i}. \end{aligned}$$

We will assume that

$$\begin{aligned} {{{\,\mathrm{{\textstyle {\sum }}}\,}}}_{ k + |J| \le 3} |\partial ^J_y \widetilde{x}_I| + |\partial ^J_y \widetilde{x}_{II}| \le M_0. \end{aligned}$$

(B.1)

By the formula for the derivative of the inverse (D.1) this implies that $|A^{\,\,\, a}_{{}_{\!}I\, i}| + |A_{{}_{\!}I{\!}I\, i}^{\,\,\,a}| \le C(M_0)$. We define

$$\begin{aligned} \widetilde{\partial }_{I i}= & {} A_{{}_{\!}I\, i}^{\,\,a} \frac{\partial }{\partial y^a},\qquad \text {and}\qquad \widetilde{\partial }_{{}_{\!}I{\!}Ii}=A_{{}_{\!}I{\!}I\, i}^{\,\,a}\frac{\partial }{\partial y^a}, \\ \widetilde{g}_{{}_{\!}I}^{ab}= & {} \delta ^{ij}A_{{}_{\!}I\, i}^{\,\,a}A_{{}_{\!}I\, j}^{\,\,b},\qquad \text {and}\qquad \widetilde{g}_{{}_{\!}I{\!}I}^{ab} = \delta ^{ij} A_{{}_{\!}I{\!}I\, i}^{\,\,\,a} A_{{}_{\!}I{\!}I\, j}^{\,\,\,b}, \end{aligned}$$

as well as:

$$\begin{aligned} \widetilde{\Delta }_{{}_{\!}I}f = \delta ^{ij}\widetilde{\partial }_{I i}\widetilde{\partial }_{I j} f = \partial _a \big (\widetilde{g}_I^{ab}\partial _b f\big ),\qquad \widetilde{\Delta }_{{}_{\!}I{\!}I}f = \delta ^{ij}\widetilde{\partial }_{I i}\widetilde{\partial }_{{}_{\!}I{\!}Ij} f = \partial _a \big (\widetilde{g}_{{}_{\!}I{\!}I}^{ab}\partial _b f\big ). \end{aligned}$$

We define

$$\begin{aligned} {{\,\mathrm{div_{I}}\,}}\alpha&= \delta ^{ij}\widetilde{\partial }_{I i} \alpha _j,&{{\,\mathrm{div_{II}}\,}}\alpha = \delta ^{ij} \widetilde{\partial }_{{}_{\!}I{\!}Ii} \alpha _j, \\ ({{\,\mathrm{curl_{I}}\,}}\alpha )_{ij}&= \widetilde{\partial }_{I i} \alpha _j - \widetilde{\partial }_{I j} \alpha _i,&({{\,\mathrm{curl_{II}}\,}}\alpha )_{ij} = \widetilde{\partial }_{{}_{\!}I{\!}Ii} \alpha _j - \widetilde{\partial }_{{}_{\!}I{\!}Ij} \alpha _i, \end{aligned}$$

and

$$\begin{aligned} \gamma _I^{ij} = A_{{}_{\!}I\, a}^{\,\,i} A_{{}_{\!}I\, b}^{\,\,j} \gamma ^{ab}, \qquad \text {and}\qquad \gamma _{{}_{\!}I{\!}I}^{ij} = A_{{}_{\!}I{\!}I\, a}^{\,\,\,i} A_{{}_{\!}I{\!}I\, b}^{\,\,\,j} \gamma ^{ab}. \end{aligned}$$

Here, we are writing $\gamma ^{ab}$ for the cometric on $\partial \Omega$ extended to the interior of $\Omega$. Fixing a smooth radial function $\chi$ with $\chi (r) = 0$ for $r \le \frac{1}{2}$ and $\chi (r) = 1$ for $r \ge \frac{3}{4}$, then

$$\begin{aligned} \gamma ^{ab} = \delta ^{ab} - \chi (r) N^a N^a, \end{aligned}$$

with N the unit normal to $\partial \Omega$.

Recalling the notation ${\mathcal {T}^{}}^r$ from Section 3.3, we will use the following norms

$$\begin{aligned}&||\alpha ||_{H^k(\Omega )}^2 =\! {\sum }_{|J| \le k} \int _\Omega \! \delta ^{ij}\partial _y^J \alpha _i \partial _y^J \alpha _j\, \mathrm{d}y,\qquad ||\alpha ||_{C^k(\Omega )} \\&\quad = \!{\sum }_{\ell = 0}^k ||\partial _y^\ell \alpha ||_{L^\infty (\Omega )},\qquad ||{\mathcal {T}^{}}^r \!\alpha ||_{L^2(\Omega )}^2 = \!\int _{\Omega } \!|{\mathcal {T}^{}}^r\! \alpha |^2 \mathrm{d}y. \end{aligned}$$

In what follows we will use the convention that the components of $\alpha$ will be expressed in terms of the $\,\widetilde{\!x}_{{}_{\!}I}$ frame and $\beta$ will be expressed in terms of the $\,\widetilde{\!x}_{{}_{\!}I\!I}$ frame and we will just write $\alpha , \beta$ instead of $\alpha _I, \beta _{II}$. We now list the elliptic estimates we use. Proofs can be found in the following sections:

Lemma B.1

With the above definitions, if $\alpha , \beta$ are (0,1)-tensors on $\Omega$, then on $[0 ,T] \times \Omega$,

$$\begin{aligned}&|\widetilde{\partial }_{{}_{\!}I}\alpha - \widetilde{\partial }_{{}_{\!}I{\!}I}\beta | \le C(M')\big ( |\!{{\,\mathrm{div_{I}}\,}}\alpha - {{\,\mathrm{div_{II}}\,}}\beta |+ |\!{{\,\mathrm{curl_{I}}\,}}\alpha - {{\,\mathrm{curl_{II}}\,}}\beta | \nonumber \\&\quad + |{\mathcal {T}^{}}\!\alpha - {\mathcal {T}^{}}\beta | + ||\,\widetilde{\!x}_{{}_{\!}I}- \,\widetilde{\!x}_{{}_{\!}I\!I}||_{C^{1}(\Omega )} |\widetilde{\partial }_{{}_{\!}I{\!}I}\beta |\big ). \end{aligned}$$

(B.2)

There is a higher-order version of Lemma B.1 in Sobolev spaces and with mixed space and time derivatives:

Lemma B.2

Fix $r \ge 7$ and let $1 \le \ell \le r$. Suppose $\,\widetilde{\!x}_{{}_{\!}I}, \,\widetilde{\!x}_{{}_{\!}I\!I}\in H^r(\Omega )$ satisfy (B.1). If $\alpha - \beta \in H^\ell _{loc}(\Omega )$ and

$$\begin{aligned}&{{\,\mathrm{div_{I}}\,}}\alpha - {{\,\mathrm{div_{II}}\,}}\beta ,\,\, {{\,\mathrm{curl_{I}}\,}}\alpha - {{\,\mathrm{curl_{II}}\,}}\beta \,\, \in H^{\ell -1}(\Omega ),\\&\quad T (\alpha \!-\! \beta ) \in H^{\ell -1}(\Omega ), \text { for all } T \!\in {\mathcal {T}^{}},\\&\quad \widetilde{\partial }_{{}_{\!}I{\!}I}\beta \in H^{\ell -1}(\Omega ), \end{aligned}$$

then $\alpha - \beta \in H^{\ell }(\Omega )$ and there is a constant $C_r = C_r(M_0, ||\,\widetilde{\!x}_{{}_{\!}I}||_{H^{r}(\Omega )}, ||\,\widetilde{\!x}_{{}_{\!}I\!I}||_{H^{r}(\Omega )})$, so that

$$\begin{aligned}&||\alpha - \beta ||_{H^{\ell }(\Omega )} \le C_r \big ( ||{{\,\mathrm{div_{I}}\,}}\alpha - {{\,\mathrm{div_{II}}\,}}\beta ||_{H^{\ell -1}(\Omega )} + ||{{\,\mathrm{curl_{I}}\,}}\alpha - {{\,\mathrm{curl_{II}}\,}}\beta ||_{H^{\ell -1}(\Omega )}\nonumber \\&\quad + ||{\mathcal {T}^{}}^{\ell -1} (\widetilde{\partial }_{{}_{\!}I}\alpha - \widetilde{\partial }_{{}_{\!}I{\!}I}\beta )||_{L^2(\Omega )} + (||\,\widetilde{\!x}_{{}_{\!}I}- \,\widetilde{\!x}_{{}_{\!}I\!I}||_{C^2(\Omega )} \nonumber \\&\quad + ||\,\widetilde{\!x}_{{}_{\!}I}- \,\widetilde{\!x}_{{}_{\!}I\!I}||_{H^{\ell }(\Omega )}) ||\widetilde{\partial }_{{}_{\!}I{\!}I}\beta ||_{H^{\ell }(\Omega )}\big ). \end{aligned}$$

(B.3)

Similarly, if $k +\ell = s \le r$, $D_t^{k'}\widetilde{\partial }\beta \in H^{\ell '}(\Omega )$ for any $k'+\ell '\!\le \! s$ and

$$\begin{aligned}&D_t^k ({{\,\mathrm{div_{I}}\,}}\!\alpha - {{\,\mathrm{div_{II}}\,}}\beta )\!\in \! H^{\ell \!-\!1\!}(\Omega ),\qquad D_t^k({{\,\mathrm{curl_{I}}\,}}\! \alpha - {{\,\mathrm{curl_{II}}\,}}\beta ) \! \in \! H^{\ell \!-\!1\!}(\Omega ),\\&\qquad D_t^k T (\alpha - \beta ) \!\in \! H^{\ell \!-\!1\!}(\Omega ), \!\!\!\!\quad \text {for all } T \!\in \! {\mathcal {T}^{}}\!\!, \end{aligned}$$

then $D_t^k(\alpha - \beta ) \in H^\ell (\Omega )$ and there is a constant $C_r' = C_r'(M_0, ||\,\widetilde{\!x}_{{}_{\!}I}||_r, ||\,\widetilde{\!x}_{{}_{\!}I\!I}||_r)$, so that

$$\begin{aligned}&||\alpha - \beta ||_{k,\ell } \le C_s' \big ( || ({{\,\mathrm{div_{I}}\,}}\alpha - {{\,\mathrm{div_{II}}\,}}\beta )||_{k,\ell -1} \nonumber \\&\quad + ||{{\,\mathrm{curl_{I}}\,}}\alpha - {{\,\mathrm{curl_{II}}\,}}\beta ||_{k,\ell -1}\nonumber \\&\quad + ||{\mathfrak {D}}^{k,\ell -1} (\widetilde{\partial }_{{}_{\!}I}\alpha - \widetilde{\partial }_{{}_{\!}I{\!}I}\beta )||_{L^2}+ ||\alpha - \beta ||_{s,0} \nonumber \\&\quad +(||\,\widetilde{\!x}_{{}_{\!}I}- \,\widetilde{\!x}_{{}_{\!}I\!I}||_{C^2(\Omega )} + ||\,\widetilde{\!x}_{{}_{\!}I}- \,\widetilde{\!x}_{{}_{\!}I\!I}||_{s}) (||\beta ||_{s,0} + ||\widetilde{\partial }\beta ||_{s-1}) \big ). \end{aligned}$$

(B.4)

In the special case that $\alpha = \partial f, \beta = \partial g$ for functions $f, g \in H^1_0(\Omega )$, $\widetilde{\partial }_{{}_{\!}I}\! f - \widetilde{\partial }_{{}_{\!}I{\!}I}g \in H^\ell _{loc}(\Omega )$, we have

Proposition B.3

Suppose $\widetilde{x}_I, \widetilde{x}_{{}_{\!}I{\!}I}\! \in \! H^s(\Omega )$, $s\! \ge \! 1$, satisfy (B.1), $f\!-_{\!}g \!\in \!H^1_0(\Omega )$, $\widetilde{\partial }_{{}_{\!}I}\! f \!-_{\!} \widetilde{\partial }_{{}_{\!}I{\!}I}g \!\in \! H^s_{loc}(\Omega )$ and that:

$$\begin{aligned} \widetilde{\Delta }_{{}_{\!}I}f - \widetilde{\Delta }_{{}_{\!}I{\!}I}g \in H^{s-1}(\Omega ),&\widetilde{\partial }_{{}_{\!}I{\!}I}g \in H^{s}(\Omega ),&T^J (\widetilde{\partial }_{{}_{\!}I}f - \widetilde{\partial }_{{}_{\!}I{\!}I}g) \in L^2(\Omega ), \text { for all } |J| \le s. \end{aligned}$$

Then $\widetilde{\partial }_{{}_{\!}I}f\! - \widetilde{\partial }_{{}_{\!}I{\!}I}g \!\in \! H^{s}(\Omega )$ and there is a constant $C_s\! = \!C_s(M_0, ||\,\widetilde{\!x}_{{}_{\!}I}||_{H^s(\Omega )}, ||\,\widetilde{\!x}_{{}_{\!}I\!I}||_s)$ so that

$$\begin{aligned}&||\widetilde{\partial }_{{}_{\!}I}f - \widetilde{\partial }_{{}_{\!}I{\!}I}g||_{H^{s}(\Omega )} \le C_s ||\widetilde{\Delta }_{{}_{\!}I}f - \widetilde{\Delta }_{{}_{\!}I{\!}I}g||_{H^{s-1}(\Omega )} +C_s ||{\mathcal {T}^{}}(\,\widetilde{\!x}_{{}_{\!}I}- \,\widetilde{\!x}_{{}_{\!}I\!I})||_{H^s(\Omega )} ||\widetilde{\partial }_{{}_{\!}I{\!}I}g||_{H^s(\Omega )}\\&\quad + C_s ||{\mathcal {T}^{}}\,\widetilde{\!x}_{{}_{\!}I}||_{H^s(\Omega )} \big (||\,\widetilde{\!x}_{{}_{\!}I}- \,\widetilde{\!x}_{{}_{\!}I\!I}||_{H^s(\Omega )} ||\widetilde{\partial }_{{}_{\!}I{\!}I}g||_{H^{s-1}(\Omega )} + ||f - g||_{L^2}\big ). \end{aligned}$$

Similarly, if $k + \ell = s$, the assumption (D.7) holds, $D_t^k(\widetilde{\partial }_{{}_{\!}I}f - \widetilde{\partial }_{{}_{\!}I{\!}I}g) \in H^{\ell }_{loc}(\Omega )$ and

$$\begin{aligned}&D_t^k(\widetilde{\Delta }_{{}_{\!}I}f- \widetilde{\Delta }_{{}_{\!}I{\!}I}g) \in H^{\ell -1}(\Omega ), \qquad D_t^k \widetilde{\partial }_{{}_{\!}I{\!}I}g \in H^{\ell }(\Omega ),\\&\qquad T^J(\widetilde{\partial }_{{}_{\!}I}f - \widetilde{\partial }_{{}_{\!}I{\!}I}g) \in L^2(\Omega ), \text { for all } T^J \in {\mathfrak {D}}^s, \end{aligned}$$

then $D_t^k(\widetilde{\partial }_{{}_{\!}I}f - \widetilde{\partial }_{{}_{\!}I{\!}I}g ) \in H^\ell (\Omega )$ and there are constants $C_s' = C_s'(M, ||\,\widetilde{\!x}_{{}_{\!}I}||_{s}, ||\,\widetilde{\!x}_{{}_{\!}I\!I}||_{s})$ so that if $k + \ell = s$,

$$\begin{aligned}&||\widetilde{\partial }_{{}_{\!}I}f - \widetilde{\partial }_{{}_{\!}I{\!}I}g||_{k,\ell } \le C_s' \big ( ||\widetilde{\Delta }_{{}_{\!}I}f - \widetilde{\Delta }_{{}_{\!}I{\!}I}g||_{k-1, \ell } \nonumber \\&\quad + ||f - g||_{s+1,0} + ||\widetilde{\partial }_{{}_{\!}I}f - \widetilde{\partial }_{{}_{\!}I{\!}I}g||_{s-1,1} + ||{\mathcal {T}^{}}\,\widetilde{\!x}_{{}_{\!}I\!I}||_s ||f - g||_{s}\big )\nonumber \\&\quad + C_r'||{\mathcal {T}^{}}(\,\widetilde{\!x}_{{}_{\!}I}- \,\widetilde{\!x}_{{}_{\!}I\!I})||_{s}\big ( ||\widetilde{\partial }_{{}_{\!}I{\!}I}g||_{s} + ||g||_{s+1,0}). \end{aligned}$$

(B.5)

We also need a result to build regularity for a function f with $\widetilde{\Delta }f \in H^{\ell -1}(\Omega )$ but with a priori only $f \in H^1_0(\Omega )$. Note that we are not assuming that $f \in H^\ell _{loc}(\Omega )$. This result is needed to prove a local-wellposedness result for the wave equation (4.6)–(4.7) (see Appendix F.1). Writing $\widetilde{x}= \widetilde{x}_I$, we have

Proposition B.4

Suppose $\widetilde{x}\in H^r(\Omega )$, $r\! \ge \! 5$, satisfies (B.1). If $f \in H^1_0(\Omega )$ and $\widetilde{\Delta }f \in H^{\ell -1}(\Omega )$ for some $0 \le \ell \le r$, then $\widetilde{\partial }f \in H^\ell (\Omega )$ and

$$\begin{aligned} ||\widetilde{\partial }f||_{H^\ell (\Omega )} \le C(M_0, ||\widetilde{x}||_{H^r(\Omega )}) \big ( ||\widetilde{\Delta }f||_{H^{\ell -1}(\Omega )} + ||{\mathcal {T}^{}}\widetilde{x}||_{H^r(\Omega )} ||f||_{L^2(\Omega )}\big ). \end{aligned}$$

Similarly, if $f \in H^1_0(\Omega )$, $D_t^k f \in L^2(\Omega )$ and $D_t^k \widetilde{\Delta }f \in H^{\ell -1}(\Omega )$, then $D_t^k \widetilde{\partial }f \in H^\ell (\Omega )$ and

$$\begin{aligned} ||D_t^k \widetilde{\partial }f||_{H^\ell (\Omega )} \le C(M_0, ||\widetilde{x}||_r) \big ( ||D_t^k \widetilde{\Delta }f||_{H^{\ell -1}(\Omega )} + ||{\mathcal {T}^{}}\widetilde{x}||_{r} ||D_t^k f||_{L^2(\Omega )}\big ). \end{aligned}$$

We also need estimates which involve fractional derivatives on $\partial \Omega$.

Proposition B.5

Let $\alpha$ be a vector field on $\Omega$. Fix $r \ge 5$. Then, for $1 \le \ell \le r$, there are continuous functions $C_\ell = C_\ell \big (M_0, ||\widetilde{x}||_{H^r(\Omega )}\big )$ so that

$$\begin{aligned}&||\alpha ||_{H^{\ell {}_{\!}}}^2 \!\le \! C_{\ell }\Big (||\!{{\,\mathrm{div}\,}}\alpha ||_{H^{\ell \!-\!1\!}}^2\! + ||\!{{\,\mathrm{curl}\,}}\alpha ||_{H^{\ell \!-\!1\!}}^2 +\!||\alpha ||^2_{H^1} \nonumber \\&\quad +\!{\sum }_{\mu = 1}^N \!\int _{\partial \Omega } \!\!({}_{\!}\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu {\mathcal {T}^{}}^{\ell -\!1 \!}\alpha ^i)\! \cdot \!(\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu {\mathcal {T}^{}}^{\ell -\!1 \!}\alpha ^j) N_{{}_{\!}i} N_{\!j} \mathrm{d}S \Big ), \end{aligned}$$

(B.6)

$$\begin{aligned}&||\alpha ||_{H^{\ell {}_{\!}}}^2 \le C_{\ell }\Big ( ||\!{{\,\mathrm{div}\,}}\alpha ||_{H^{\ell \!-\!1\!}}^2\! + ||\!{{\,\mathrm{curl}\,}}\alpha ||_{H^{\ell \!-\!1\!}}^2\! +\!||\alpha ||^2_{H^1} \nonumber \\&\quad + {\sum }_{\mu = 1}^N \!\int _{\partial \Omega } \!\!(\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu {\mathcal {T}^{}}^{\ell -1 \!}\alpha ^i)\!\cdot \!(\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu {\mathcal {T}^{}}^{\ell -1\!} \alpha ^j) \gamma _{ij} \mathrm{d}S\Big ). \end{aligned}$$

(B.7)

We will need the following lemma to exchange normal and tangential components of vector fields on $\partial \Omega$. This estimate appears in Lemma 5.6 of [3].

Lemma B.6

If $\alpha$ is a (0,1)-tensor on $\Omega$ and $\gamma$ denotes the metric on $\partial \Omega _t$, then

$$\begin{aligned} \Big |\int _{\partial \Omega } \big (\gamma ^{ij} - N^iN^j\big )\alpha _i \alpha _j d\mu _\gamma \Big | \le \big ( ||{{\,\mathrm{div}\,}}\alpha ||_{L^2(\Omega )} + ||{{\,\mathrm{curl}\,}}\alpha ||_{L^2(\Omega )} + K ||\alpha ||_{L^2(\Omega )}\big ) ||\alpha ||_{L^2(\Omega )}. \end{aligned}$$

Finally, in Section F.1, we will need the following elliptic estimate in $H^2(\Omega )$:

Lemma B.7

Let $\Delta _y\! = \partial _{{}_{\!}y^1}^2\! + \partial _{{}_{\!}y^2}^2\! + \partial _{{}_{\!}y^3}^2$ be the flat Laplacian in the y coordinates. If $f \!\in \! H^{{}_{\!}1}_0(\Omega ) \!\cap \! H^2(\Omega )$, then:

$$\begin{aligned} ||\widetilde{\partial }f||_{H^1(\Omega )} \le C(M) \big ( (\Delta _y f, \widetilde{\Delta }f)_{L^2(\Omega )} + ||\widetilde{\partial }f||_{L^2(\Omega )} + ||f||_{L^2(\Omega )}\big ). \end{aligned}$$

(B.8)

1.1 Proof of Lemma B.1

The case with $\beta = 0$ is Lemma 5.5 in [3], and this version is Lemma B.4.1 of [21]. For the reader’s convenience, we include the proof here. We start by setting

$$\begin{aligned} \big ({{\,\mathrm{def}\,}}_I \alpha \big )_{ij} = \widetilde{\partial }_{I{}_{\!}i} \alpha _{I{}_{\!}j} + \widetilde{\partial }_{I{}_{\!}j} \alpha _{I{}_{\!}i},\quad (D_I \alpha )_{ij} = {{\,\mathrm{div}\,}}_I \alpha \delta _{ij},\quad (\widehat{D_I} \alpha )_{ij} = \big ({{\,\mathrm{def}\,}}_I \alpha - \frac{2}{3}D_I\alpha \big )_{ij}, \end{aligned}$$

with a similar definition for ${{\,\mathrm{def}\,}}_{{}_{\!}I{\!}I}, D_{{}_{\!}I{\!}I},$ and $\widehat{D_{{}_{\!}I{\!}I}}$. We write

$$\begin{aligned} \widetilde{\partial }_{{}_{\!}I}\alpha - \widetilde{\partial }_{{}_{\!}I{\!}I}\beta&= \frac{1}{3}\big (D_I \alpha - D_{{}_{\!}I{\!}I} \beta \big ) + \frac{1}{2}\big ({{\,\mathrm{curl}\,}}_I \alpha - {{\,\mathrm{curl}\,}}_{{}_{\!}I{\!}I} \beta \big ) + \frac{1}{2}\big (\widehat{D_{I}} \alpha - \widehat{D_{{}_{\!}I{\!}I}} \beta \big ). \end{aligned}$$

The first and second terms are bounded by the right-hand side of (B.2), and we now show how to control the last term. Let $S_{ij} = (\widehat{D_I}\alpha - \widehat{D_{II}} \beta )_{ij}$. Writing $\delta ^{ij} = \gamma _I^{ij} - N^i_I N^j_I$ and using that S is symmetric, we have

$$\begin{aligned} \delta ^{ij}\delta ^{k\ell } S_{ik}S_{j\ell } = \big (\gamma _I^{ij}\gamma _{I}^{k\ell } + 2 \gamma _I^{ij} N_I^{k\ell } + N_I^iN_I^jN_I^kN^\ell _I\big ) S_{ik} S_{j\ell }. \end{aligned}$$

(B.9)

Now, because $\delta ^{ij} S_{ij} = 0$, the last term is:

$$\begin{aligned} \big (N_I^iN_I^j S_{ij}\big )^2 = \big (\delta ^{ij}S_{ij} - \gamma _I^{ij}S_{ij} \big )^2 = \big (\gamma _I^{ij} S_{ij}\big )^2 \le 2 \gamma _I^{ij} \gamma _I^{k\ell } S_{ik}S_{j\ell }, \end{aligned}$$

where we have used that if $T\!$ is a symmetric matrix then $({{\,\mathrm{tr}\,}}T)^2\! {}_{\!}\le \! \text {rank} T {}_{\!}{{\,\mathrm{tr}\,}}(T^2)$. Returning to (B.9), we have

$$\begin{aligned} |S|^2 \le 2 \gamma _I^{ij}\big (\gamma _I^{k\ell } + N_I^k N_I^\ell \big ) S_{ik}S_{j\ell } = 2\gamma _I^{ij} \delta ^{k\ell }S_{ik}S_{j\ell }. \end{aligned}$$

We now write

$$\begin{aligned} S_{ij} = \big ( {{\,\mathrm{def}\,}}_I \alpha - {{\,\mathrm{def}\,}}_{II}\beta \big )_{ij} - \frac{2}{3} \big ( D_I\alpha - D_{II}\beta \big )_{ij} \equiv S^1_{ij} + S^2_{ij}. \end{aligned}$$

Since $|S^2| \le C |{{\,\mathrm{div}\,}}_I \alpha - {{\,\mathrm{div}\,}}_{II}\beta |$, it suffices to control $S^1$. We have

$$\begin{aligned}&\gamma _I^{ij}\delta ^{k\ell }S^1_{ik}S^1_{j\ell } = \gamma _I^{ij}\delta ^{k\ell } \big ( \widetilde{\partial }_{I i} \alpha _{I k} - \widetilde{\partial }_{{}_{\!}I{\!}Ii} \beta _{I\!I k } \nonumber \\&\quad + \widetilde{\partial }_{I k} \alpha _{I i} - \widetilde{\partial }_{{}_{\!}I{\!}Ik} \beta _{I\!I i} \big ) \big ( \widetilde{\partial }_{I j}\alpha _{I\ell } - \widetilde{\partial }_{{}_{\!}I{\!}Ij}\beta _{II\ell } \nonumber \\&\quad + \widetilde{\partial }_{I \ell }\alpha _{Ij} - \widetilde{\partial }_{{}_{\!}I{\!}I\ell }\beta _{IIj}\big ) . \end{aligned}$$

(B.10)

To bound the product of the first term in the first factor with the first term in the second factor, we replace $\widetilde{\partial }_{II}\beta _{II}$ with $\widetilde{\partial }_{I} \beta _{II}$, which generates terms that are bounded by the last term on the right-hand side of (B.2). The resulting term only involves tangential derivatives of $\alpha , \beta$ but these are with respect to $\,\widetilde{\!x}_{{}_{\!}I}$. However we can replace these with tangential derivatives with respect to y up to terms that are bounded by the last term on the right-hand side of (B.2). For the product of the second term in the first factor and the second term in the second factor we instead note that it can be controlled in terms of $|\!{{\,\mathrm{curl}\,}}_I\!\alpha \!- {{\,\mathrm{curl}\,}}_{II}\! \beta |^2$ along with the third and fourth terms on the right-hand side of (B.2) The other terms in (B.10) can be handled similarly.

1.2 Proof of Lemma B.2

Both estimates have essentially the same proof, so we will just prove the second. The first one follows from the same argument, but one uses the commutator estimate D.5 with $\mathcal {V}\!=\! \{\partial _{y^1\!}, \partial _{y^2\!}, \partial _{y^3}\!\}$ instead of $\mathcal {V}\!=\! \mathcal {D}$. The only difference is that in the proof of (B.3) no time derivatives enter.

We argue by induction. When $s = 1$, the result follows from the pointwise estimate after writing

$$\begin{aligned} \partial _{a} (\alpha - \beta )= A_{{}_{\!}I\, a}^{\,\,i} ( \widetilde{\partial }_{I{}_{\!}i}\alpha - \widetilde{\partial }_{{}_{\!}I{\!}I{}_{\!}i}\beta ) + (A_{{}_{\!}I\, a}^{\,\,i} - A_{{}_{\!}I{\!}I\, a}^{\,\,\,i})\widetilde{\partial }_{I{}_{\!}i}\beta . \end{aligned}$$

We now assume that we have the result for $s \!\le \! m\!-\!1$. We write $T^I\!\! = D_t^k \partial _y^J \!\!\in \mathcal {D}^{k,\ell }$ where $k \!+ \! |J|\! = m$. If $|J|\!=\! 0$ there is nothing to prove, so we consider $|J| \!\ge \! 1$. We then write $D_t^k \partial _y^J \!\!= \partial _{a} D_t^k \partial _y^{J'}\!$ where $J \!= (a, J')$ and $\partial _{a} \!= A_{{}_{\!}I\, a}^{\,\,i} \widetilde{\partial }_{i}$. Applying the pointwise estimate (B.2) and integrating over an arbitrary $U \!\subset \subset \Omega$, we have

$$\begin{aligned}&||T^I (\alpha - \beta )||_{L^2(U)} \le C(M_0) \big ( ||{{\,\mathrm{div_{I}}\,}}D_t^k \partial _y^{J'} \alpha - {{\,\mathrm{div_{II}}\,}}D_t^k \partial _y^{J'} \beta ||_{L^2(\Omega )} \nonumber \\&\quad + ||{{\,\mathrm{curl_{I}}\,}}D_t^k \partial _y^{J'} \alpha - {{\,\mathrm{curl_{II}}\,}}D_t^k \partial _y^{J'} \beta ||_{L^2(\Omega )} \nonumber \\&\quad + ||{\mathcal {T}^{}}D_t^k \partial _y^{J'} (\alpha - \beta )||_{L^2(\Omega )} \nonumber \\&\quad + ||\,\widetilde{\!x}_{{}_{\!}I}- \,\widetilde{\!x}_{{}_{\!}I\!I}||_{C^2(\Omega )} ||\widetilde{\partial }_{{}_{\!}I{\!}I}D_t^k \partial _y^{J'}\beta ||_{L^2(\Omega )}\big ). \end{aligned}$$

(B.11)

Using the commutator estimate from Lemma D.5 with $\mathcal {V}= \mathcal {D}$, the last term is bounded by the right-hand side of (B.4). To deal with the first two terms, we apply the commutator estimate (D.9) with $\mathcal {V}= \mathcal {D}$ to set

$$\begin{aligned}&||{{\,\mathrm{div_{I}}\,}}\! D_t^k \partial _y^{J'}\!\alpha - {{\,\mathrm{div_{II}}\,}}\! D_t^k\partial _y^{J'} \!\beta ||_{L^2} \le ||D_t^k \partial _y^{J'}\! ({{\,\mathrm{div_{I}}\,}}\! \alpha - {{\,\mathrm{div_{II}}\,}}\beta )||_{L^2} \nonumber \\&\quad + C_s \big ( ||\widetilde{\partial }_{{}_{\!}I}\alpha - \widetilde{\partial }_{{}_{\!}I{\!}I}\beta ||_{m-2} + ||\,\widetilde{\!x}_{{}_{\!}I}\!- \,\widetilde{\!x}_{{}_{\!}I\!I}||_{s} ||\widetilde{\partial }_{{}_{\!}I{\!}I}\beta ||_{m-2}\big ), \end{aligned}$$

where $L^2=L^2(\Omega )$ and $C_s = C_s(M, ||\,\widetilde{\!x}_{{}_{\!}I}||_s, ||\,\widetilde{\!x}_{{}_{\!}I\!I}||_s)$, along with a similar estimate for the curl. All of these terms are bounded by the right-hand side of (B.4). To deal with the last term on the right-hand side of (B.11), we commute the tangential derivative with $D_t^k\partial _y^{J'}$ to set

$$\begin{aligned} |{\mathcal {T}^{}}D_t^k \partial _y^{J'} (\alpha - \beta )| \le {\sum }_{T \in {\mathcal {T}^{}}} |D_t^k \partial _y^{J'} T(\alpha - \beta )| + C |D_t^k\partial _y^{J'} (\alpha - \beta )|. \end{aligned}$$

The second term here is bounded by the right-hand side of (B.4) by the inductive assumption. To control the first term in $L^2$, we apply the inductive assumption with $\alpha , \beta$ replaced by $T \alpha , T\beta$, and this gives

$$\begin{aligned}&||D_t^k\partial _y^{J'} T(\alpha - \beta )||_{L^2(\Omega )} \le C_s ||{{\,\mathrm{div_{I}}\,}}T\alpha - {{\,\mathrm{div_{II}}\,}}T \beta ||_{k,\ell -2} \nonumber \\&\quad + ||{{\,\mathrm{curl_{I}}\,}}T\alpha - {{\,\mathrm{curl_{II}}\,}}T \beta ||_{k,\ell -2} + ||{\mathfrak {D}}^{k,\ell }(\alpha - \beta )||_{L^2(\Omega )}\nonumber \\&\quad + C_s(||\,\widetilde{\!x}_{{}_{\!}I}- \,\widetilde{\!x}_{{}_{\!}I\!I}||_{C^2(\Omega )} + ||\,\widetilde{\!x}_{{}_{\!}I}-\,\widetilde{\!x}_{{}_{\!}I\!I}||_{r} )||\widetilde{\partial }_{{}_{\!}I{\!}I}T\beta ||_{m-1}. \end{aligned}$$

(B.12)

We now write ${{\,\mathrm{div}\,}}T (\alpha \!-\! \beta ) = T {{\,\mathrm{div}\,}}(\alpha \! -\! \beta ) - TA_{{}_{\!}I\, i}^{\,\,a} \partial _a (\alpha ^i\! -\! \beta ^i)$, and use the product rule (A.25) and (D.1):

$$\begin{aligned} ||(TA_{{}_{\!}I\, i}^{\,\,a})\partial _a(\alpha ^i - \beta ^i)||_{k,\ell -2} \le C(M, ||\,\widetilde{\!x}_{{}_{\!}I}||_s)||\alpha - \beta ||_{m-1}. \end{aligned}$$

Arguing as with the other terms in (B.12), recalling that we are integrating over any $U \!\subset \subset \!\Omega$ gives the result.

1.3 Proof of Proposition B.3

To motivate the proof, first consider the case that $\widetilde{x}_{{}_{\!}I{\!}I} \!= \!\widetilde{x}_{{}_{\!}I}$ and $g \!=\! 0$. If $\widetilde{x}_{{}_{\!}I}$ was smooth, one could get a version of this estimate without tangential derivatives by straightening the boundary and using a standard integration by parts argument. Because the coordinate $\widetilde{x}_{{}_{\!}I}$ is only smooth in tangential directions, the idea is instead to first use the estimate (B.3) to replace the derivatives of $\widetilde{\partial }f$ with derivatives of $\Delta f$ and tangential derivatives of $\widetilde{\partial }f$, and then apply the integration by parts argument to this. One then has to deal with commutators $[{\mathcal {T}^{}}^r\!\!, \widetilde{\partial }]f$. To highest order, this behaves like $({\mathcal {T}^{}}^r \partial _y \widetilde{x}_{{}_{\!}I})\partial _y f$, and because the derivatives ${\mathcal {T}^{}}$ are tangential this term can be handled. Also note that since ${\mathcal {T}^{}}^r\! f\!=\! 0$ on $\partial \Omega$, the boundary terms that arise when integrating by parts vanish so we avoid the need to straighten the boundary.

We start with the following estimate:

Lemma B.8

Under the hypotheses of Proposition B.3, we have

$$\begin{aligned} ||\widetilde{\partial }_{{}_{\!}I}f - \widetilde{\partial }_{{}_{\!}I{\!}I}g||_{L^2}^2 \le C(M_0)\big ( ||\widetilde{\Delta }_{{}_{\!}I}f - \widetilde{\Delta }_{{}_{\!}I{\!}I}g||_{L^2}^2 + ||\,\widetilde{\!x}_{{}_{\!}I}- \,\widetilde{\!x}_{{}_{\!}I\!I}||_{C^2(\Omega )}^2 ||\widetilde{\partial }_{{}_{\!}I{\!}I}g||_{L^2}^2\big ). \end{aligned}$$

(B.13)

Proof

We write $\widetilde{\partial }_{{}_{\!}I{\!}I}g = \widetilde{\partial }_{{}_{\!}I}g + (A_{{}_{\!}I{\!}I}-A_I)\cdot \partial _y g$ and since $||\alpha ||_{L^2(\Omega )}^2$ is comparable to $\int _\Omega |\alpha |^2 \widetilde{\kappa }\, \mathrm{d}y$, so

$$\begin{aligned}&||\widetilde{\partial }_{{}_{\!}I}f - \widetilde{\partial }_{{}_{\!}I{\!}I}g||_{L^2(\Omega )}^2 \lesssim \int _{\Omega } \delta ^{ij} \big (\widetilde{\partial }_{I i} f - \widetilde{\partial }_{{}_{\!}I{\!}Ii} g\big ) \big ( \widetilde{\partial }_{I j} f - \widetilde{\partial }_{{}_{\!}I{\!}Ij}g\big )\widetilde{\kappa }_{I}\mathrm{d}y \\&\quad = \int _{\Omega } \delta ^{ij} \big ( \widetilde{\partial }_{I i} f - \widetilde{\partial }_{{}_{\!}I{\!}Ii} g\big ) \big ( \widetilde{\partial }_{I j} f - \widetilde{\partial }_{{}_{\!}I{\!}Ij}g\big )\widetilde{\kappa }_{I}\mathrm{d}y\\&\qquad + 2\int _{\Omega } \delta ^{ij} (A_{{}_{\!}I\, i}^{\,\,a} - A_{{}_{\!}I{\!}I\, i}^{\,\,\,a}) A_{{}_{\!}I\, j}^{\,\,b} (\partial _a g)\partial _b (f-g)\widetilde{\kappa }_{I}\, \mathrm{d}y \\&\qquad + \int _{\Omega } \delta ^{ij} (A_{{}_{\!}I\, i}^{\,\,a} - A_{{}_{\!}I{\!}I\, i}^{\,\,\,a}) (A_{{}_{\!}I\, j}^{\,\,b} - A_{{}_{\!}I{\!}I\, j}^{\,\,\,b}) (\partial _a g)(\partial _b g) \widetilde{\kappa }_{I}\mathrm{d}y. \end{aligned}$$

The terms on the last line are bounded by the second term on the right-hand side of (B.13), using Lemma D.2 and Sobolev embedding. To control the terms on the first line, we integrate by parts to set

$$\begin{aligned}&\int _\Omega \delta ^{ij} A_{{}_{\!}I\, i}^{\,\,a} A_{\,\,i}^a \partial _a (f-g) A_{{}_{\!}I\, j}^{\,\,b}\partial _b(f-g) \widetilde{\kappa }_{I}\mathrm{d}y \\&\quad = -\int _\Omega (f-g) \frac{1}{\widetilde{\kappa }_{I}} \partial _a \big ( \widetilde{\kappa }_{I}\delta ^{ij} A_{{}_{\!}I\, i}^{\,\,a}A_{{}_{\!}I\, j}^{\,\,b} \partial _b (f-g)\big ) \widetilde{\kappa }_{I}\mathrm{d}y. \end{aligned}$$

The second factor here is $\widetilde{\Delta }_{{}_{\!}I}(f-g) = \big (\widetilde{\Delta }_{{}_{\!}I}f - \widetilde{\Delta }_{{}_{\!}I{\!}I}g \big )+ (\widetilde{\Delta }_{{}_{\!}I}- \widetilde{\Delta }_{{}_{\!}I{\!}I})g$. Since we want a bound that only involves one derivative of g, we further write

$$\begin{aligned} (\widetilde{\Delta }_{{}_{\!}I}- \widetilde{\Delta }_{{}_{\!}I{\!}I})g = \frac{1}{\widetilde{\kappa }_{I}} \partial _a \Big ( \widetilde{\kappa }_{I}(\widetilde{g}_{{}_{\!}I}^{ab} \partial _b g) - \widetilde{\kappa }_{{}_{\!}I{\!}I}(\widetilde{g}_{{}_{\!}I{\!}I}^{ab}\partial _b g) \Big ) + \Big (\frac{1}{\widetilde{\kappa }_{{}_{\!}I{\!}I}} - \frac{1}{\widetilde{\kappa }_{I}}\Big ) \partial _a \big ( \widetilde{\kappa }_{{}_{\!}I{\!}I}\widetilde{g}_{{}_{\!}I{\!}I}^{ab}\partial _b g\big ), \end{aligned}$$

and then integrate by parts and use Poincarè’s inequality again, which shows that

$$\begin{aligned} \Big | \int _\Omega (f-g) (\widetilde{\Delta }_{{}_{\!}I}- \widetilde{\Delta }_{{}_{\!}I{\!}I}) g\widetilde{\kappa }\mathrm{d}y\Big | \le C(M_0)||\,\widetilde{\!x}_{{}_{\!}I}-\,\widetilde{\!x}_{{}_{\!}I\!I}||_{C^2(\Omega )} ||\widetilde{\partial }_{{}_{\!}I}f - \widetilde{\partial }_{{}_{\!}I{\!}I}g||_{L^2} ||\widetilde{\partial }_{{}_{\!}I{\!}I}g||_{L^2}. \end{aligned}$$

$\square$

We now consider the case $\alpha = \widetilde{\partial }_{{}_{\!}I}f, \beta = \widetilde{\partial }_{{}_{\!}I{\!}I}g$ for functions $f,g \in H^1_0(\Omega )$. We then have

Proposition B.9

With the hypotheses of Proposition B.3, for each s there are constants$C_s = C_s(M, ||\,\widetilde{\!x}_{{}_{\!}I}||_{H^s(\Omega )}, ||D_t \,\widetilde{\!x}_{{}_{\!}I}||_{s}, ||\,\widetilde{\!x}_{{}_{\!}I\!I}||_{H^s(\Omega )}, ||D_t \,\widetilde{\!x}_{{}_{\!}I\!I}||_s)$ so that if $k + \ell = s$,

$$\begin{aligned}&||\widetilde{\partial }_{{}_{\!}I}f - \widetilde{\partial }_{{}_{\!}I{\!}I}g||_{k,\ell } \le C_s\big ( ||\widetilde{\Delta }_{{}_{\!}I}f - \widetilde{\Delta }_{{}_{\!}I{\!}I}g||_{k-1, \ell } \\&\quad + ||f - g||_{s,0} + ||\widetilde{\partial }_{{}_{\!}I}f - \widetilde{\partial }_{{}_{\!}I{\!}I}g||_{s-1,1} + ||{\mathcal {T}^{}}\,\widetilde{\!x}_{{}_{\!}I\!I}||_s ||f - g||_{s}\\&\quad + ||{\mathcal {T}^{}}(\,\widetilde{\!x}_{{}_{\!}I}- \,\widetilde{\!x}_{{}_{\!}I\!I})||_{s}\big ( ||\widetilde{\partial }_{{}_{\!}I{\!}I}g||_{s} + ||g||_{s+1,0}\big )\big ). \end{aligned}$$

This proposition follows from (B.4) and the following lemma:

Lemma B.10

With the hypotheses as above, there is a constant $C_{\!s}(M,_{\!} ||\,\widetilde{\!x}_{{}_{\!}I}||_{s},_{\!} ||\,\widetilde{\!x}_{{}_{\!}I\!I}||_{s})$ so that, for any $\delta \! > \!0$,

$$\begin{aligned}&||{\mathfrak {D}}^{k,\ell } (\widetilde{\partial }_{{}_{\!}I}f - \widetilde{\partial }_{{}_{\!}I{\!}I}g)||_{L^2(\Omega )} \le C_s \big (||\widetilde{\Delta }_{{}_{\!}I}f- \widetilde{\Delta }_{{}_{\!}I{\!}I}g||_{k-1,\ell } + \delta ||\widetilde{\partial }_{{}_{\!}I}f \nonumber \\&\quad - \widetilde{\partial }_{{}_{\!}I{\!}I}g||_{k,\ell }+ \delta ^{-1}||{\mathcal {T}^{}}(\,\widetilde{\!x}_{{}_{\!}I}-\,\widetilde{\!x}_{{}_{\!}I\!I})||_s ||\widetilde{\partial }_{{}_{\!}I{\!}I}g||_{s}\nonumber \\&\quad + \delta ^{-1}||{\mathcal {T}^{}}\,\widetilde{\!x}_{{}_{\!}I}||_s (||\widetilde{\partial }_{{}_{\!}I}f - \widetilde{\partial }_{{}_{\!}I{\!}I}g||_{s,0} + ||f - g||_{s,0}) \big ),\quad s\!=\!k\!+\!\ell . \end{aligned}$$

(B.14)

Proof of Lemma B.10

For the purposes of the below proof, the commutator $[T, \partial _a]$ for $T \in {\mathcal {T}^{}}$ will be ignored for notational convenience. We argue by induction. When $s = 1$, we fix a multi-index I with $|I| = 1$. If $T^I = D_t$ there is nothing to prove so we assume that $T^I = S \in {\mathcal {T}^{}}$. We start by writing

$$\begin{aligned}&||S \widetilde{\partial }_{{}_{\!}I}f\! - S \widetilde{\partial }_{{}_{\!}I{\!}I}g||_{L^2(\Omega )}^2 = \int _\Omega \!\!(\widetilde{\partial }_{{}_{\!}I}Sf\! - \widetilde{\partial }_{{}_{\!}I{\!}I}Sg)\cdot S( \widetilde{\partial }_{{}_{\!}I}f \!- \widetilde{\partial }_{{}_{\!}I{\!}I}g)\,\mathrm{d}y \\&\quad + \!\int _\Omega \!\! \big ([\widetilde{\partial }_{{}_{\!}I}, S] f \!- [\widetilde{\partial }_{{}_{\!}I{\!}I}, S] g\big ) \cdot S (\widetilde{\partial }_{{}_{\!}I}f \!- S \widetilde{\partial }_{{}_{\!}I{\!}I}g)\, \mathrm{d}y. \end{aligned}$$

To deal with the first term, we integrate by parts and use that $Sf \!=\! Sg\!=\! 0$ on $\partial \Omega$, which gives

$$\begin{aligned}&\int _\Omega \!\!(\widetilde{\partial }_{{}_{\!}I}S f - \widetilde{\partial }_{{}_{\!}I{\!}I}Sg)\!\cdot \! S (\widetilde{\partial }_{{}_{\!}I}{}_{\!} f \!- \widetilde{\partial }_{{}_{\!}I{\!}I}g) \mathrm{d}y\\&\quad = \!\int _\Omega \!\! Sf \partial _a\big ( \delta ^{ij}\! A_{{}_{\!}I\, i}^{\,\,a} \{ S \widetilde{\partial }_{{}_{\!}I}{}_{\!j} f \!- S\widetilde{\partial }_{{}_{\!}I{\!}I}{}_{\!j} g\}\big ) \mathrm{d}y \\&\quad - \!\int _\Omega \!\! Sg\, \partial _a\big ( \delta ^{ij}\! A_{{}_{\!}I{\!}I\, i}^{\,\,\,a} \{S \widetilde{\partial }_{{}_{\!}I}{}_{\!j} f \!- S\widetilde{\partial }_{{}_{\!}I{\!}I}{}_{\!j} g\}\big ) \mathrm{d}y. \end{aligned}$$

We write the first term on the right-hand side as

$$\begin{aligned}&\int _\Omega \delta ^{ij} \big ( (Sf)A_{{}_{\!}I\, i}^{\,\,a} - (Sg) A_{{}_{\!}I{\!}I\, i}^{\,\,\,a}) \partial _a \big ( S \widetilde{\partial }_{{}_{\!}I}{}_{\!j}f - S\widetilde{\partial }_{{}_{\!}I{\!}I}{}_{\!j} g\big )\, \mathrm{d}y \\&\quad +\int _\Omega \delta ^{ij} \big (Sf \partial _a( A_{{}_{\!}I\, i}^{\,\,a}) - S g\partial _a( A_{{}_{\!}I{\!}I\, i}^{\,\,\,a})\big ) \big (S\widetilde{\partial }_{{}_{\!}I}f - S\widetilde{\partial }_{{}_{\!}I{\!}I}g\big )\, \mathrm{d}y. \end{aligned}$$

The second term here is bounded by the right-hand side of (B.14). We now re-write the first term as

$$\begin{aligned}&\int _\Omega \!\! (Sf) \widetilde{\partial }_{{}_{\!}I}{}_{\!}\cdot {}_{\!} (S \widetilde{\partial }_{{}_{\!}I}f\! - S\widetilde{\partial }_{{}_{\!}I{\!}I}g) - (Sg) \widetilde{\partial }_{{}_{\!}I{\!}I}{}_{\!} \cdot {}_{\!} (S \widetilde{\partial }_{{}_{\!}I}f\! - S \widetilde{\partial }_{{}_{\!}I{\!}I}g) \\&\quad =\!\! \int _\Omega \!(Sf) S (\widetilde{\Delta }_{{}_{\!}I}f\! - \widetilde{\partial }_{{}_{\!}I}\!\cdot {}_{\!} \widetilde{\partial }_{{}_{\!}I{\!}I}g) - (Sg) S(\widetilde{\partial }_{{}_{\!}I{\!}I}\!\cdot \widetilde{\partial }_{{}_{\!}I}{}_{\!}f \!- \widetilde{\Delta }_{{}_{\!}I{\!}I}g)\\&\quad +\!\! \int _\Omega \!(S{}_{\!}f) [\widetilde{\partial }_{{}_{\!}I}, S] {}_{\!}\cdot {}_{\!} ( \widetilde{\partial }_{{}_{\!}I}{}_{\!}f \!- \widetilde{\partial }_{{}_{\!}I{\!}I}g) \\&\quad -(Sg) [\widetilde{\partial }_{{}_{\!}I{\!}I}, S] {}_{\!}\cdot {}_{\!} (\widetilde{\partial }_{{}_{\!}I}{}_{\!}f \! - \widetilde{\partial }_{{}_{\!}I{\!}I}g). \end{aligned}$$

Finally, we re-write the first term on the right-hand side as

$$\begin{aligned} \int _\Omega (Sf - Sg) S (\widetilde{\Delta }_{{}_{\!}I}f - \widetilde{\Delta }_{{}_{\!}I{\!}I}g) + \int _\Omega (Sf - Sg) S (\widetilde{\partial }_{{}_{\!}I}- \widetilde{\partial }_{{}_{\!}I{\!}I})\cdot \widetilde{\partial }_{{}_{\!}I{\!}I}g, \end{aligned}$$

and integrate S by parts in each of these terms. Applying Cauchy’s inequality, the result of the above is

$$\begin{aligned}&||S(\widetilde{\partial }_{{}_{\!}I}f - \widetilde{\partial }_{{}_{\!}I{\!}I}g)||_{L^2(\Omega )}^2 \le C_1 \big (||\widetilde{\Delta }_{{}_{\!}I}f - \widetilde{\Delta }_{{}_{\!}I{\!}I}g||_{L^2(\Omega )}^2 \\&\quad + \delta ^{-1}||\widetilde{\partial }_{{}_{\!}I}f - \widetilde{\partial }_{{}_{\!}I{\!}I}g||_{L^2(\Omega )}^2\big )+ C_1 \delta ^{-1}||\,\widetilde{\!x}_{{}_{\!}I}- \,\widetilde{\!x}_{{}_{\!}I\!I}||_{C^2(\Omega )} ||\widetilde{\partial }_{{}_{\!}I{\!}I}g||_{H^1(\Omega )}\\&\quad + C_1\delta \big (||\widetilde{\partial }_{{}_{\!}I}f - \widetilde{\partial }_{{}_{\!}I{\!}I}g||_{H^1(\Omega )}^2 + ||[\widetilde{\partial }_{{}_{\!}I}, S] f \\&\quad - [\widetilde{\partial }_{{}_{\!}I{\!}I}, S] g||_{L^2(\Omega )}^2 + ||[\widetilde{\partial }_{{}_{\!}I}, S]\cdot \widetilde{\partial }_{{}_{\!}I}f - [\widetilde{\partial }_{{}_{\!}I{\!}I}, S]\cdot \widetilde{\partial }_{{}_{\!}I{\!}I}g||_{L^2(\Omega )}^2\big ). \end{aligned}$$

Here, and in what follows, we will use $C_k$ to denote a constant which depends on $M, ||\,\widetilde{\!x}_{{}_{\!}I}||_k, ||\,\widetilde{\!x}_{{}_{\!}I\!I}||_k$. Applying the commutator estimate (D.9), every term here is bounded by the right-hand side of (B.14).

We now suppose we have the result for $s = 1,..., m-1$, and fix $T^I = D_t^k T^J$ where $T^K \in {\mathcal {T}^{}}^\ell$ with $k + \ell = m$. If $|K| = 0$ there is nothing to prove so we assume that $T^I = ST^J$ for some $S \in {\mathcal {T}^{}}$ and $T^J \in {\mathfrak {D}}^{k,\ell -1}$. The proof now follows in nearly the same way as above, so we just indicate the main points. First, we write

$$\begin{aligned}&\int _\Omega \!\! T^I \!(\widetilde{\partial }_{{}_{\!}I}{}_{\!}f \!- \widetilde{\partial }_{{}_{\!}I{\!}I}g) T^I\! (\widetilde{\partial }_{{}_{\!}I}{}_{\!} f\! - \widetilde{\partial }_{{}_{\!}I{\!}I}g)\, \mathrm{d}y = \int _\Omega \!\! (\widetilde{\partial }_{{}_{\!}I}T^I\! {}_{\!}f\! - \widetilde{\partial }_{{}_{\!}I{\!}I}T^I \! g) T^I\!(\widetilde{\partial }_{{}_{\!}I}{}_{\!}f\! - \widetilde{\partial }_{{}_{\!}I{\!}I}g) \mathrm{d}y \\&\quad +\! \int _\Omega \!\! ([\widetilde{\partial }_{{}_{\!}I},\! T^I]f\! - [\widetilde{\partial }_{{}_{\!}I{\!}I},\! T^I] g) T^I (\widetilde{\partial }_{{}_{\!}I}{}_{\!}f\!- \widetilde{\partial }_{{}_{\!}I{\!}I}g) \mathrm{d}y. \end{aligned}$$

Integrating by parts in the first term yields, in addition to lower-order terms, we have

$$\begin{aligned} \int _\Omega (T^I f)\widetilde{\partial }_{{}_{\!}I}\cdot (T^I \widetilde{\partial }_{{}_{\!}I}f - T^I \widetilde{\partial }_{{}_{\!}I{\!}I}g) - (T^I g) \widetilde{\partial }_{{}_{\!}I{\!}I}\cdot (T^I \widetilde{\partial }_{{}_{\!}I}f - T^I \widetilde{\partial }_{{}_{\!}I{\!}I}g)\, \mathrm{d}y. \end{aligned}$$

We now write $T^I = S T^J$ in the second factor in each term and then commute S with $\widetilde{\partial }_{{}_{\!}I}, \widetilde{\partial }_{{}_{\!}I{\!}I}$, and obtain

$$\begin{aligned} \int _\Omega (T^J f - T^J g) S(T^J\widetilde{\Delta }_{{}_{\!}I}f - T^J \widetilde{\Delta }_{{}_{\!}I{\!}I}g) + \int _\Omega (T^J f - T^J g) S T^J\big ( (\widetilde{\partial }_{{}_{\!}I}- \widetilde{\partial }_{{}_{\!}I{\!}I}) \cdot \widetilde{\partial }_{{}_{\!}I{\!}I}g\big ). \end{aligned}$$

Integrating S by parts and bounding

$$\begin{aligned} ||T^J\big ((\widetilde{\partial }_{{}_{\!}I}- \widetilde{\partial }_{{}_{\!}I{\!}I}) \cdot \widetilde{\partial }_{{}_{\!}I{\!}I}g\big )||_{L^2(\Omega )} \le ||\,\widetilde{\!x}_{{}_{\!}I}- \,\widetilde{\!x}_{{}_{\!}I\!I}||_{r} ||\widetilde{\partial }_{{}_{\!}I{\!}I}g||_{m} \end{aligned}$$

shows that $||T^I (\widetilde{\partial }_{{}_{\!}I}f - \widetilde{\partial }_{{}_{\!}I{\!}I}g)||_{L^2(\Omega )}$ is bounded by

$$\begin{aligned}&C_m \big ( ||T^J (\widetilde{\Delta }_{{}_{\!}I}f - \widetilde{\Delta }_{{}_{\!}I{\!}I}g)||_{L^2(\Omega )} + (1 + \delta ^{-1})||T^J(\widetilde{\partial }_{{}_{\!}I}f - \widetilde{\partial }_{{}_{\!}I{\!}I}g)||_{L^2(\Omega )}^2\\&\quad + (1 + \delta ^{-1}) ||{\mathcal {T}^{}}(\,\widetilde{\!x}_{{}_{\!}I}- \,\widetilde{\!x}_{{}_{\!}I\!I})||_{m}^2 ||\widetilde{\partial }g||_{k,\ell }^2\\&\quad + C_m \delta \big ( ||\widetilde{\partial }_{{}_{\!}I}f - \widetilde{\partial }_{{}_{\!}I{\!}I}g||_{k,\ell }^2 + || [\widetilde{\partial }_{{}_{\!}I}, S] T^J f - [\widetilde{\partial }_{{}_{\!}I{\!}I}, S] T^J g||_{L^2(\Omega )} \\&\quad + ||[\widetilde{\partial }_{{}_{\!}I}, S] T^J \widetilde{\partial }_{{}_{\!}I}f - [\widetilde{\partial }_{{}_{\!}I}, S] T^J \widetilde{\partial }_{{}_{\!}I{\!}I}g||_{L^2(\Omega )} \big ). \end{aligned}$$

The result now follows after using the commutator estimate (D.9) and induction.

$\square$

1.4 Proof of Proposition B.4

We just prove the $k = 0$ case, as the $k \ge 1$ case follows using similar arguments. This would be a consequence of the Proposition B.3 with $g = 0$ if we knew that $\widetilde{\partial }f \in H^m_{loc}(\Omega )$ and ${\mathcal {T}^{}}^I \widetilde{\partial }f\in L^2(\Omega )$ for all $|I| \le m$. In the following lemma we prove that this is the case (see Section A for the definitions of the sets $U_\alpha$ and the vector fields $T \in {\mathcal {T}^{}}$):

Lemma B.11

Fix $s \!\ge \! 0$ and suppose that $\widetilde{x}\!\in \! H^s(\Omega ), T \!\widetilde{x}\! \in \! H^s(\Omega )$ for all $T \!\in \!{\mathcal {T}^{}}\!$ and that (B.1) holds. Suppose also that $f\! {}_{\!}\in \! H^{s{}_{\!}}(\Omega )$, $\widetilde{\Delta }f\!{}_{\!} \in \! H^{s-\!1\!}(\Omega )$. Then $\widetilde{\partial }{}_{\!}f\! {}_{\!}\in \! H^s_{loc}(\Omega )$ and $T^{I{}_{\!}} \widetilde{\partial }{}_{\!}f {}_{\!}\!\in \! L^2(\Omega )$ for all $|I|\! \le \!s$ and there is a constant $C_s \!=\! C_s(M_0, ||\widetilde{x}||_{H^s(\Omega )})$ so that with notation as in (3.5), the following inequalities hold for any $V\! \subset \subset \!\Omega$:

$$\begin{aligned}&||\widetilde{\partial }f||_{H^s(V)} + ||{\mathcal {T}^{}}^s\widetilde{\partial }f||_{L^2(\Omega )} \le C_s\big ( ||\widetilde{\Delta }f||_{H^{s-1}(\Omega )}\nonumber \\&\quad + (||{\mathcal {T}^{}}\widetilde{x}||_{H^s(\Omega )} + ||\widetilde{x}||_{H^s(\Omega )}) ||f||_{H^s(\Omega )}\big ). \end{aligned}$$

(B.15)

Proof

We will follow the proof in [9]. Both of the above statements have essentially the same proof and so we will just prove the second one. For the case $s = 1$, we want to show that

$$\begin{aligned} {\sum }_{T \in {\mathcal {T}^{}}} ||T \widetilde{\partial }f||_{L^2(\Omega )} \le C(M')\big ( ||\widetilde{\Delta }f||_{L^2(\Omega )} + ||\widetilde{\partial }f||_{L^2(\Omega )}\big ). \end{aligned}$$

(B.16)

We fix one of the open sets $U = U_\mu$ with $\mu \ge 1$ and write $F = f\circ \psi _\mu$. Then, arguing as in [9], to prove (B.16) it suffices to prove that for every $V \subset U$, with a constant independent of h,

$$\begin{aligned} ||D_c^h \widetilde{\partial }F||_{L^2(V)} \le C \big ( ||\widetilde{\Delta }f||_{L^2(\Omega )} + ||\widetilde{\partial }f||_{L^2(\Omega )}\big ), \quad \text {for}\quad c = 1,2, \end{aligned}$$

(B.17)

for $D_{\!c}^{h}\!$ denoting the difference quotient in the direction of a unit vector $e_c$

$$\begin{aligned} D_c^h F(z) = \big (F(z + h e_c) - F(z)\big )/{h}. \end{aligned}$$

Let $\rho$ denote a cutoff function which is 1 on V and zero outside of U, and set $v = -D_c^{-h}(\rho ^2 D_c^h F)$. Note that $v \in H^1_0(U)$. Now we have

$$\begin{aligned}&\int _U \!\!\widetilde{\Delta }F v = -\!\int _U\!\!\delta ^{ij} (\widetilde{\partial }_i F) \widetilde{\partial }_j \big \{ D_c^{-h} (\rho ^2 D_c^h F)\big \} =\! \int _U\!\! \delta ^{ij}\!\underbrace{ (\widetilde{\partial }_i F) D_c^{-h} \widetilde{\partial }_j\{\rho ^2 D_c^h F\}}_{I}\\&\quad - \int _U \!\!\delta ^{ij}\! \underbrace{(\widetilde{\partial }_i F) (D_c^{-h}\! A_{\,\,j}^a) \partial _a \{\rho ^2 D_c^h F\}}_{II}. \end{aligned}$$

Now,

$$\begin{aligned}&I = \int _U (D_c^h \widetilde{\partial }_iF) \widetilde{\partial }_j (\rho ^2 D_c^h F) = \int _U \rho ^2 \delta ^{ij} (D_c^h \widetilde{\partial }_i F)(\widetilde{\partial }_j D_c^h F) \\&\qquad + \int _U 2 \delta ^{ij} (D_c^h \widetilde{\partial }_i f) (D_c^h f)(\rho \widetilde{\partial }_j \rho )\\&\quad = \int _U \rho ^2 \delta ^{ij} (D_c^h \widetilde{\partial }_i F)(D_c^h\widetilde{\partial }_j F) \\&\qquad + \int _U \delta ^{ij} (D_c^h \widetilde{\partial }_i F) \big \{ 2\rho (\widetilde{\partial }_j \rho ) (D_c^h F) - \rho ^2 (D_c^h A_{\,\,j}^a)\partial _a F)\big \}. \end{aligned}$$

The first term is

$$\begin{aligned} \int _U \rho ^2 |D_c^h \widetilde{\partial }F|^2. \end{aligned}$$

The second term is bounded by

$$\begin{aligned}&C(_{\!}M_{0\!})||\rho D_{\!c}^h {}_{\!}\widetilde{\partial }F||_{L^{\!2}(U{}_{\!})} \big (||D_{\!c}^h\! F||_{L^{\!2}(U{}_{\!})} {}_{\!}+{}_{\!} ||D_{\!c}^h \! A||_{L^{\!\infty \!}(U{}_{\!})} ||f||_{H^{1\!}}{}_{\!}\big )\! \\&\quad \le \! C(_{\!}M_{0\!}) ||\rho D_{\!c}^h {}_{\!}\widetilde{\partial }F||_{L^{\!2}(U{}_{\!})}||f||_{H^1\!(\Omega )}(1 {}_{\!}+{}_{\!} ||\partial ^2 \widetilde{x}||_{L^{\!\infty \!}}). \end{aligned}$$

Next, writing $\partial _a D_c^h F = D_c^h \partial _a F = D_c^h(A_{\,\,a}^\ell \widetilde{\partial }_\ell F)$, we have

$$\begin{aligned} II= & {} \int _U \delta ^{ij}(\widetilde{\partial }_j F)(D_c^{-h} \! A_{\,\,j}^a)\big \{ 2\rho \partial _a \rho D_c^h F + \rho ^2 \partial _a D_c^h F\big \}\\= & {} \int _U \delta ^{ij} (\widetilde{\partial }_j F)(D_c^{-h}\! A_{\,\,j}^a) \big \{ 2\rho \partial _a \rho D_c^h F + \rho ^2 A_{\,\,a}^\ell D_c^h \widetilde{\partial }_\ell F - \rho ^2 (D_c^h A_{\,\,a}^\ell )\widetilde{\partial }_\ell F\big \} \end{aligned}$$

so we have

$$\begin{aligned}&|II| \le C(M_0)||\partial ^2 \widetilde{x}||_{L^\infty (U)}||\widetilde{\partial }F||_{L^2(U)} \big (||F||_{H^1(U)} \\&\quad + ||\rho D_c^h \widetilde{\partial }F||_{L^2(U)} + ||\partial ^2 x||_{L^\infty (U)} ||\widetilde{\partial }F||_{L^2(U)}\big ). \end{aligned}$$

Finally, we have

$$\begin{aligned} \int _U |\widetilde{\Delta }F| |v|\, \mathrm{d}y \le ||\widetilde{\Delta }F||_{L^2(U)} ||D_c^{-h} (\rho ^2 D_c^h F)||_{L^2(U)}. \end{aligned}$$

Using similar arguments to the above, we can show

$$\begin{aligned} ||D_c^{-h} (\rho ^2 D_c^h F)||_{L^2(U)} \le C(M')\big ( ||\rho D_c^h \widetilde{\partial }F||_{L^2(U)} + ||\partial ^2 \widetilde{x}||_{L^\infty (U)}||F||_{H^1}\big ), \end{aligned}$$

so that

$$\begin{aligned}&\int _U \rho ^2 |D_c^h \widetilde{\partial }F|^2 \le C(M_0) \big ( ||\rho D_c^h \widetilde{\partial }F||_{L^2(U)} \big \{(1 + ||\partial ^2 x||_{L^\infty (U)})||F||_{H^1(U)} \\&\quad + ||g||_{L^2(U)}\big \} + ||f||_{H^1(U)}^2\big ). \end{aligned}$$

Absorbing this first factor into the left-hand side we have, for any h small enough,

$$\begin{aligned} \int _V |D_c^h \widetilde{\partial }F|^2 \le \int _U \rho ^2 |D_c^h \widetilde{\partial }F|^2 \le C(M_0)\big ( (1 + ||\partial ^2 \widetilde{x}||_{L^\infty (U)})^2 ||F||_{H^1(U)}^2 + ||g||_{L^2(U)}^2\big ), \end{aligned}$$

which implies the $s = 1$ case of the theorem.

Now suppose that $T^J \widetilde{\partial }F \in L^2(\Omega )$ for all $|J| \le s-1$. Fix a multi-index I with $|I| = s-1$ and write $F' = T^I F$. Note that $F' = 0$ on $\partial \Omega$ in the trace sense and also that

$$\begin{aligned} ||\partial _y F'||_{L^2(\Omega )} \le C(M_0)\widetilde{\partial }F'||_{L^2(\Omega )} \le C(M_0)\big ( ||T^I\widetilde{\partial }F||_{L^2(\Omega )} + ||[\widetilde{\partial }, {\mathcal {T}^{}}^I]F||_{L^2(\Omega )} \big ). \end{aligned}$$

The commutator can be bounded using Lemma D.5:

$$\begin{aligned} ||[\widetilde{\partial }, T^I] F||_{L^2(\Omega )} \le C(M_0, ||\widetilde{x}||_{H^s(\Omega )}) \big ( ||{\mathcal {T}^{}}\widetilde{x}||_{H^s(\Omega )} + ||\widetilde{x}||_{H^s(\Omega )} \big )||F||_{H^{s-1}(\Omega )}. \end{aligned}$$

In particular, this implies that $F' \!\!\in \! H^1_0(\Omega )$. We also have

$$\begin{aligned} \widetilde{\Delta }F' = T^I \widetilde{\Delta }F + [T^I,\widetilde{\Delta }] F, \end{aligned}$$

(B.18)

and

$$\begin{aligned} ||[T^I, \widetilde{\Delta }] F||_{L^2(\Omega )} \le C(M_0, ||\widetilde{x}||_{H^s(\Omega )}) ( ||{\mathcal {T}^{}}\widetilde{x}||_{H^s(\Omega )} + ||\widetilde{x}||_{H^s(\Omega )}) ||F||_{H^s(\Omega )}, \end{aligned}$$

Therefore we have that $F'\! \in \! H^1_0$ is the weak solution to the problem (B.18) and $\widetilde{\Delta }F'\! \in \! L^2(\Omega )$, so by the $|I| \!=\! 1$ case we have ${\mathcal {T}^{}}\! \widetilde{\partial }F' \!\in \! L^2(\Omega )$ and

$$\begin{aligned} ||T \widetilde{\partial }F'||_{L^2(\Omega )} \le C(M_0)\big ( ||\widetilde{\Delta }F'||_{L^2(\Omega )} + ||\widetilde{\partial }F'||_{L^2(\Omega )}\big ). \end{aligned}$$

(B.19)

We write

$$\begin{aligned} T \widetilde{\partial }_i F' = T (\widetilde{\partial }_i {\mathcal {T}^{}}^I F) = T T^I \widetilde{\partial }_i F + (T T^I A_{\,\,i}^a)\partial _a F + R, \end{aligned}$$

where the $L^2$ norm of R is bounded by the right side of (B.15). Combining this with (B.19) gives (B.15). To prove the first estimate in (B.15) we argue in the same way, but we also prove (B.17) also for $c \!=\! 3$. $\quad \square$

1.5 Proof of Proposition B.5

We will need a few preliminary results. First, we fix a function d with $d = 0$ on $\partial \Omega , d < 0$ in $\Omega$ and $|\nabla d| > 0$ everywhere, so that the normal can be written as

$$\begin{aligned} N_i = {\widetilde{\partial }_i d}\, /|{\widetilde{\partial }d}| = {A_{\,\,i}^a \partial _ad}\,/|\widetilde{\partial }d|,\qquad \text {where}\quad |\widetilde{\partial }d|^2=\delta ^{ij}\widetilde{\partial }_i d\, \widetilde{\partial }_j d\, =\widetilde{g}^{ab} \partial _a d\, \partial _b d. \end{aligned}$$

By (D.1) and Lemma D.1, this implies the estimates

$$\begin{aligned}&||N||_{C^\ell (\partial \Omega )} \le C(M') ||\widetilde{x}||_{C^{\ell +1}(\partial \Omega )} \le C(M')||\widetilde{x}||_{H^{\ell +4}(\Omega )},\nonumber \\&\quad \text { and } \quad ||N||_{H^{\ell }(\partial \Omega )} \le C(M') ||\widetilde{x}||_{H^{\ell +1}(\partial \Omega )}. \end{aligned}$$

(B.20)

where in the first inequality we used Sobolev embedding on $\partial \Omega$ and the trace inequality (A.21). Recalling the definition $\gamma _{ij} = \delta _{ij} - N_i N_j$, there are similar estimates for derivatives of $\gamma$.

The basic result we need is the following consequence of Green’s formula:

Lemma B.12

If $\alpha$ is a vector field, then

$$\begin{aligned}&||\widetilde{\partial }\alpha ||_{L^2(\widetilde{\mathcal {D}}_t)}^2 = ||{{\,\mathrm{div}\,}}\alpha ||_{L^2(\widetilde{\mathcal {D}}_t)}^2 + \frac{1}{2} ||{{\,\mathrm{curl}\,}}\alpha ||_{L^2(\widetilde{\mathcal {D}}_t)}^2 \\&\quad +\int _{\partial \widetilde{\mathcal {D}}_t} \Big (\alpha ^j (\gamma _j^k \widetilde{\partial }_k \alpha _i) N^i - \alpha _i (\gamma _j^k \widetilde{\partial }_k \alpha ^j )N^i\Big ). \end{aligned}$$

Proof

Integrating by parts,

$$\begin{aligned} ||\widetilde{\partial }\alpha ||_{L^2(\widetilde{\mathcal {D}}_t)}^2 = -\int _{\widetilde{\mathcal {D}}_t} \delta ^{ij} \alpha _i \widetilde{\Delta }\alpha _j + \int _{\partial \widetilde{\mathcal {D}}_t} \delta ^{ij} \alpha _i N^k \widetilde{\partial }_k \alpha _j. \end{aligned}$$

(B.21)

We insert the identity

$$\begin{aligned} \Delta \alpha _j = \delta ^{k\ell }\widetilde{\partial }_k(\widetilde{\partial }_\ell \alpha _j) = \delta ^{k\ell } \widetilde{\partial }_k \big ( \widetilde{\partial }_j \alpha _\ell + {{\,\mathrm{curl}\,}}\alpha _{\ell j}\big ) = \widetilde{\partial }_j {{\,\mathrm{div}\,}}\alpha + \delta ^{k\ell }\widetilde{\partial }_k {{\,\mathrm{curl}\,}}\alpha _{\ell j} \end{aligned}$$

into the first term in (B.21) and integrate by parts again to get

$$\begin{aligned}&\int _{\widetilde{\mathcal {D}}_t}\delta ^{ij} \alpha _i \widetilde{\Delta }\alpha _j = \int _{\partial \widetilde{\mathcal {D}}_t} N^i \alpha _i {{\,\mathrm{div}\,}}\alpha + \delta ^{ij} N^\ell \alpha _i {{\,\mathrm{curl}\,}}\alpha _{\ell j} \mathrm{d}S \\&\quad - \int _{\widetilde{\mathcal {D}}_t} ({{\,\mathrm{div}\,}}\alpha )^2 + \delta ^{k\ell }\delta ^{ij} \widetilde{\partial }_k \alpha _i {{\,\mathrm{curl}\,}}\alpha _{\ell j}. \end{aligned}$$

Note that by the antisymmetry of curl,

$$\begin{aligned}&\delta ^{k\ell }\delta ^{ij} \widetilde{\partial }_k \alpha _i {{\,\mathrm{curl}\,}}\alpha _{\ell j}\!=\frac{1}{2} \delta ^{k\ell }\delta ^{ij} (\widetilde{\partial }_k \alpha _i+\widetilde{\partial }_i \alpha _k) {{\,\mathrm{curl}\,}}\alpha _{\ell j}\\&\quad +\frac{1}{2} \delta ^{k\ell }\delta ^{ij} (\widetilde{\partial }_k \alpha _i-\widetilde{\partial }_i \alpha _k) {{\,\mathrm{curl}\,}}\alpha _{\ell j}\!=\frac{1}{2} \delta ^{k\ell }\delta ^{ij} {{\,\mathrm{curl}\,}}\alpha _{k i} {{\,\mathrm{curl}\,}}\alpha _{\ell j}, \end{aligned}$$

so (B.21) becomes

$$\begin{aligned}&||\widetilde{\partial }\alpha ||_{L^2(\widetilde{\mathcal {D}}_t)}^2 = ||{{\,\mathrm{div}\,}}\alpha ||_{L^2(\widetilde{\mathcal {D}}_t)}^2 + \frac{1}{2} ||{{\,\mathrm{curl}\,}}\alpha ||_{L^2(\widetilde{\mathcal {D}}_t)}^2 +\int _{\partial \widetilde{\mathcal {D}}_t} N^k \alpha ^j \widetilde{\partial }_k \alpha _j \\&\quad - N^i \alpha _i {{\,\mathrm{div}\,}}\alpha - N^\ell \alpha ^j {{\,\mathrm{curl}\,}}\alpha _{\ell j} . \end{aligned}$$

Here

$$\begin{aligned}&N^k \alpha ^j \widetilde{\partial }_k \alpha _j - N^i \alpha _i {{\,\mathrm{div}\,}}\alpha - N^\ell \alpha ^j {{\,\mathrm{curl}\,}}\alpha _{\ell j}= N^k \alpha ^j \widetilde{\partial }_j \alpha _k - N^i \alpha _i {{\,\mathrm{div}\,}}\alpha \\&\quad =N^k \alpha _\ell N^\ell N^j\widetilde{\partial }_j \alpha _k +N^k \alpha _\ell \gamma ^{\ell j}\widetilde{\partial }_j \alpha _k - N^i \alpha _i( N^k N^\ell + \gamma ^{\ell k})\widetilde{\partial }_k \alpha _\ell \\&\quad =N^k \alpha _\ell \gamma ^{\ell j}\widetilde{\partial }_j \alpha _k - N^i \alpha _i \gamma ^{\ell k}\widetilde{\partial }_k \alpha _\ell . \end{aligned}$$

$\square$

Lemma B.13

There is a constant $C_{\!1}$ depending on $M'\!\!$ and $||\widetilde{x}||_{H^5(\Omega )}$ so that if $\alpha$ is a vector field on $\Omega$ then

$$\begin{aligned}&\! ||\alpha ||_{H^{1{}_{\!}}(\Omega )}^2 {}_{\!}\le \! C_{{}_{\!}1\!}\Big (\! ||\!{{\,\mathrm{div}\,}}\alpha ||_{L^2(\Omega )}^2\! +{}_{\!} ||\!{{\,\mathrm{curl}\,}}\alpha ||_{L^2(\Omega )}^2\! \nonumber \\&\quad + {\sum }_{\mu =1}^N\!\int _{\partial \Omega }\!\!\! \big (\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu \alpha ^i\big ) \big (\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu \alpha ^j\big ) N_i N_j \mathrm{d}S + ||\alpha ||_{L^2(\partial \Omega )}^2\! + ||\alpha ||_{L^2(\Omega )}^2\!\Big ) , \nonumber \\ \end{aligned}$$

(B.22)

$$\begin{aligned}&||\alpha ||_{H^{1{}_{\!}}(\Omega )}^2 \!\le \! C_1 \Big ( ||\!{{\,\mathrm{div}\,}}\alpha ||_{L^2(\Omega )}^2\! + ||\!{{\,\mathrm{curl}\,}}\alpha ||_{L^2(\Omega )}^2 \nonumber \\&\quad +{\sum }_{\mu =1}^N\!\int _{\partial \Omega } \!\!\!(\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu \alpha ^i)(\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu \alpha ^j) \gamma _{ij} \mathrm{d}S + ||\alpha ||_{L^2(\partial \Omega )}^2\! + ||\alpha ||_{L^2(\Omega )}^2\!\Big ). \end{aligned}$$

(B.23)

Proof

These estimates follow from the fact that, for any $\epsilon > 0$,

$$\begin{aligned}&\big |\int _{\partial \Omega }\!\!\!\!\! \alpha ^j (\gamma _j^k \widetilde{\partial }_k \alpha _i) N^i \!\!- \alpha _i(\gamma _j^k \widetilde{\partial }_k \alpha ^j)N^i\, \mathrm{d}S \big |\nonumber \\&\quad \le C(M', ||\widetilde{x}||_{H^5(\Omega )})\Big ( \frac{1}{\epsilon } {\sum }_{\mu =1}^N||(\langle \partial _{\theta {}_{\!}} \rangle ^{1/2}\alpha ) \cdot N||_{L^2(\partial \Omega )}^2 \nonumber \\&\quad + \epsilon {\sum }_{\mu =1}^N||(\langle \partial _{\theta {}_{\!}} \rangle ^{1/2}\alpha )\cdot \gamma ||_{L^2(\partial \Omega )}^2 + ||\alpha ||_{L^2(\Omega )}^2\Big ). \end{aligned}$$

(B.24)

To see that this estimate implies (B.22), we use (B.20) and the trace inequality (A.21) to control the second term by $C(M') ||\alpha ||_{H^{1/2}(\partial \Omega )} \le C(M')||\alpha ||_{H^1(\Omega )}$, and then take $\epsilon$ sufficiently small. The estimate (B.23) follows by instead using the trace estimate on the first term and taking $\epsilon$ sufficiently large.

To prove (B.24), we write $\gamma _j^k \widetilde{\partial }_k \alpha ^j\!\! = \gamma _j^k \widetilde{\partial }_k(\gamma ^j_\ell \alpha ^\ell ) - \gamma ^k_j (\widetilde{\partial }_k\gamma ^j_\ell )\alpha ^\ell -\gamma _j^k (\widetilde{\partial }_k N^j) N_{\!\ell }\alpha ^\ell$ and the left-hand side as

$$\begin{aligned}&\int _{\partial \Omega } \alpha ^j \gamma _j^k\widetilde{\partial }_k(\alpha _i N^i) - \gamma _j^k (\widetilde{\partial }_k(\gamma ^j_\ell \alpha ^\ell ) ) \alpha _iN^i \\&\quad + \int _{\partial \Omega } \gamma ^k_j (\widetilde{\partial }_k\gamma ^j_\ell )\alpha ^\ell \alpha _i N^i +\gamma _j^k (\widetilde{\partial }_k N^j) N_\ell \alpha ^\ell \alpha _i N^i -\alpha ^j \alpha _i \gamma _j^k \widetilde{\partial }_k N^i. \end{aligned}$$

The second integral is bounded by the right-hand side of (B.24), by (B.20). The first integral is bounded by the right-hand side of (B.24) using the fractional product rules (A.3) and (A.4) - (A.6). $\quad \square$

Proof of Proposition B.5

By the previous lemma we have the result for $\ell = 1$. Assume that we have the result for $\ell = 1, ..., m-1$. To prove it for $\ell = m$, we write $\partial _y^m \alpha = \partial _y^{m-1} \widetilde{\partial }\alpha + [\partial ^{m-1},\widetilde{\partial }]\alpha$. This second term can be bounded by the third term on the right-hand side of (B.6) (resp. (B.7)) by using Lemma D.1 and arguing as in the proof of Proposition B.2. To control the first term, we apply (B.3) and we need to control $||\widetilde{\partial }{{\,\mathrm{div}\,}}\alpha ||_{H^{m-2}(\Omega )},||\widetilde{\partial }{{\,\mathrm{curl}\,}}\alpha ||_{H^{m-2}(\Omega )}$ and $||{\mathcal {T}^{}}^J \widetilde{\partial }\alpha ||_{L^2(\Omega )}$ for all multi-indices with $|J| = m-1$. Writing $\widetilde{\partial }= A\cdot \partial _y$ and arguing as above, the first two terms are bounded by the right-hand side of (B.6) (resp. (B.7)). It therefore just remains to control the third term. We commute $T^J$ with $\widetilde{\partial }$, apply (D.3) and again argue as in the proof of Proposition B.2. Applying (B.22) (resp. (B.23)) and repeating the same argument as above completes the proof of Proposition B.5. $\square$

Proof of Lemma B.7

It suffices to prove the claim for $f \in C_c^\infty (\Omega )$ by an approximation argument. Integrating by parts twice and using that $\partial _a \widetilde{\partial }_i = \widetilde{\partial }_i \partial _a -(\partial _a A_{\,\,i}^c)\partial _c$, we have

$$\begin{aligned}&(\widetilde{\Delta }f, \Delta f)_{L^2(\Omega )} = \int _\Omega \delta ^{ij} \delta ^{ab} (\widetilde{\partial }_i \widetilde{\partial }_j f)(\partial _a \partial _b f)\\&\quad = \int _\Omega \delta ^{ij} \delta ^{ab} (\partial _a \widetilde{\partial }_j f)(\partial _b \widetilde{\partial }_i f) \\&\quad + \int _\Omega \delta ^{ij} \delta ^{ab} (\widetilde{\partial }_j f)(\partial _a A_{\,\,j}^c)(\partial _c \partial _b f) -\int _\Omega \delta ^{ij} \delta ^{ab} (\partial _a \widetilde{\partial }_j f)(\partial _b A_{\,\,i}^d)(\partial _d f). \end{aligned}$$

This implies that

$$\begin{aligned} (\widetilde{\Delta }f, \Delta f)_{L^2(\Omega )} \ge C(M) \big ( ||\widetilde{\partial }f||_{H^1(\Omega )}^2 - ||\widetilde{\partial }f||_{H^1(\Omega )} ||f||_{H^1(\Omega )}\big ), \end{aligned}$$

and the result follows. $\quad \square$

Appendix C: Proofs of Elliptic Estimates for the Newton Potential

In this section we record the elliptic estimates that are needed to control $\phi$ in Section 7. We will use the convention in (7.1) for functions $C_s, C_s^\prime , C_s^{\prime \prime }, C_s^{\prime \prime \prime }$ throughout this section.

1.1 C.1: Estimates for Section 7.1

Let $\widehat{\mathcal {D}}_t$ be the extended fluid domain (see Section 7) and $\widehat{\partial }$ be the associated spatial derivative.

Lemma C.1

Suppose $r\ge 5$. If $\widehat{\Delta }f=g$ in $\widehat{\mathcal {D}}_t$, then for $j\le r-1$,

$$\begin{aligned}&||\widehat{\partial }{\mathcal {T}^{}}^j \widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}\le C_r \bigg ({\sum }_{k\le j+1}||{\mathcal {T}^{}}^{k}\widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}\!+{\sum }_{k\le 2}||{\mathcal {T}^{}}^k g||_{L^6(\widehat{\mathcal {D}}_t)} \nonumber \\&\quad +||{\mathcal {T}^{}}^j\! g||_{L^2(\widehat{\mathcal {D}}_t)}+||g||_{L^\infty (\widehat{\mathcal {D}}_t)}\bigg ). \end{aligned}$$

(C.1)

If $j\! \le \! r\!-\!2$ then (C.1) holds without the $L^6\!$ norms, and if $j \! \le r\!-1$ it holds without the $L^\infty$ norm. In addition,

$$\begin{aligned}&\!\!\! ||\widehat{\partial }{\mathcal {T}^{}}^r \widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}\le C_r \bigg ({\sum }_{k\le r+1}||{\mathcal {T}^{}}^{k}\widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}\!+||{\mathcal {T}^{}}^r\! g||_{L^2(\widehat{\mathcal {D}}_t)}\nonumber \\&\quad +||{\mathcal {T}^{}}\widetilde{x}||_{H^r(\Omega )}\big [{\sum }_{k\le 4}||{\mathcal {T}^{}}^k \widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}+{\sum }_{k\le 2}||{\mathcal {T}^{}}^k g||_{L^6(\widehat{\mathcal {D}}_t)}+||g||_{L^\infty (\widehat{\mathcal {D}}_t)}\big ]\bigg ). \nonumber \\ \end{aligned}$$

(C.2)

Moreover, for $0\le \ell \le 2$,

$$\begin{aligned} ||\widehat{\partial }{\mathcal {T}^{}}^{\ell } \widehat{\partial }f||_{L^6(\widehat{\mathcal {D}}_t)} \le C_r\bigg ({\sum }_{k\le \ell }||{\mathcal {T}^{}}^k g||_{L^6(\widehat{\mathcal {D}}_t)}+{\sum }_{k\le \ell +2}||{\mathcal {T}^{}}^k \widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}\bigg ), \end{aligned}$$

(C.3)

as well as

$$\begin{aligned} ||\widehat{\partial }^2 f||_{L^\infty (\widehat{\mathcal {D}}_t)}\le C_0\big (||g||_{L^\infty (\widehat{\mathcal {D}}_t)}+{\sum }_{|J|\le 1}||\widehat{\partial }^J{\mathcal {T}^{}}\widehat{\partial }f||_{L^6(\widehat{\mathcal {D}}_t)}\big ). \end{aligned}$$

(C.4)

The above estimates also hold in the domain $\widetilde{\mathcal {D}}_t$ with $\widetilde{\partial }$ instead of $\widehat{\partial }$.

Proof

The estimate (C.4) follows from the pointwise estimate (5.5) and Sobolev embedding:

$$\begin{aligned}&||\widehat{\partial }^2 f||_{L^\infty (\widehat{\mathcal {D}}_t)} \le C_0\big ( ||g||_{L^\infty (\widehat{\mathcal {D}}_t)}+||{\mathcal {T}^{}}\widehat{\partial }f||_{L^\infty (\widehat{\mathcal {D}}_t)} \big )\\&\quad \le C_0\big ( ||g||_{L^\infty (\widehat{\mathcal {D}}_t)}+{\sum }_{|J|\le 1}||\widehat{\partial }^J {\mathcal {T}^{}}\widehat{\partial }f||_{L^6(\widehat{\mathcal {D}}_t)}\big ). \end{aligned}$$

By (5.5) we also have that

$$\begin{aligned}&||\widehat{\partial }{\mathcal {T}^{}}^{\ell } \widehat{\partial }f||_{L^6(\widehat{\mathcal {D}}_t)} \le C_0\big (||{{\,\mathrm{div}\,}}{\mathcal {T}^{}}^{\ell } \widehat{\partial }f||_{L^6(\widehat{\mathcal {D}}_t)}\\&\quad +||{{\,\mathrm{curl}\,}}{\mathcal {T}^{}}^{\ell } \widehat{\partial }f||_{L^6(\widehat{\mathcal {D}}_t)}+||{\mathcal {T}^{}}^{1+\ell }\widehat{\partial }f||_{L^6(\widehat{\mathcal {D}}_t)} \big )\\&\quad \le C_0\big (||{\mathcal {T}^{}}^{\ell } g||_{L^6(\widehat{\mathcal {D}}_t)} +{\sum }_{k_1+k_2=\ell -1}||({\mathcal {T}^{}}^{1+k_1} \widehat{A})(\widehat{\partial }{\mathcal {T}^{}}^{k_2}\widehat{\partial }f)||_{L^6(\widehat{\mathcal {D}}_t)}\\&\quad +|| {\mathcal {T}^{}}^{\ell +1}\widehat{\partial }f||_{H^1(\widehat{\mathcal {D}}_t)} \big ), \end{aligned}$$

where the sum is not there if $\ell \! =\! 0$. Putting ${\mathcal {T}^{}}^{1+k_1}\!\! \widehat{A}$ into ${L}^{\!\infty }\!$ and using induction, this implies that for $\ell \!\le \! 2$,

$$\begin{aligned} ||\widehat{\partial }{\mathcal {T}^{}}^{\ell } \widehat{\partial }f||_{L^6(\widehat{\mathcal {D}}_t)} \le C_0\big (||{\mathcal {T}^{}}^{\ell } g||_{L^6(\widehat{\mathcal {D}}_t)}+{\sum }_{\ell ' \le \ell } ||{\mathcal {T}^{}}^{\ell '+1}\widehat{\partial }f||_{H^1(\widehat{\mathcal {D}}_t)} \big ). \end{aligned}$$

(C.5)

We now prove (C.1), which, combined with (C.5) will also prove (C.3). We proceed by induction: for $j\!=\!0$, (C.2) without the $L^6$ and $L^\infty$ norms is a direct consequence of (5.5). Now suppose that (C.2) is known for $j=0,1,\cdots , m-1 \le r-1$. Using the pointwise estimate (5.5) we have

$$\begin{aligned}&||\widehat{\partial }{\mathcal {T}^{}}^m \widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}\le C_0\big (||{{\,\mathrm{div}\,}}{\mathcal {T}^{}}^m \widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}\\&\quad +||{{\,\mathrm{curl}\,}}{\mathcal {T}^{}}^m \widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)} +||{\mathcal {T}^{}}^{m+1}\widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}\big ). \end{aligned}$$

Here ${{\,\mathrm{div}\,}}$ and ${{\,\mathrm{curl}\,}}$ stand for the divergence and curl with respect to $\widehat{\partial }$. Since ${{\,\mathrm{div}\,}}{\mathcal {T}^{}}^m \widehat{\partial }_{\!} f\!=\!{\mathcal {T}^{}}^m \!g+\sum ({\mathcal {T}^{}}^k \!\!\widehat{A})\widehat{\partial }{\mathcal {T}^{}}^\ell \widehat{\partial }_{\!} f$, where $\widehat{A}=(\widehat{A}_{\,\,i}^a)$ and the sum is over $k+\ell =m$ with $k\ge 1$, we have

$$\begin{aligned} {{\,\mathrm{div}\,}}{\mathcal {T}^{}}^m \widehat{\partial }f = {\mathcal {T}^{}}^m g+\sum ({\mathcal {T}^{}}^{k_1}\partial \widehat{x})\cdots ({\mathcal {T}^{}}^{k_s}\partial \widehat{x})(\widehat{\partial }{\mathcal {T}^{}}^\ell \widehat{\partial }f). \end{aligned}$$

(C.6)

The above sum is over $k_1+\cdots +k_s+\ell =k+\ell =m, k \ge 1$ , and $\partial$ denotes the Lagrangian spatial derivative $\partial _y$. This is because ${\mathcal {T}^{}}^k\widehat{A}$ is a sum of terms of the form $({\mathcal {T}^{}}^{k_1}\partial \widehat{x})\cdots ({\mathcal {T}^{}}^{k_s}\partial \widehat{x})$. Now, we need to control $\sum ({\mathcal {T}^{}}^{k_1}\partial \widehat{x})\cdots ({\mathcal {T}^{}}^{k_s}\partial \widehat{x})(\widehat{\partial }{\mathcal {T}^{}}^\ell \widehat{\partial }f)$ in $L^{\!2}(\widehat{\mathcal {D}}_{t\!})$. When $\ell \!\ge \! 3$, then $k_1,\cdots \!, k_s\!\le \! r\!-\!3$, so all terms involving $\widehat{x}$ can be controlled in $L^{\!\infty }$ by $||\widehat{x}||_{H^r(\Omega ^{d_0})}$ and we control $||\widehat{\partial }{\mathcal {T}^{}}^\ell \widehat{\partial }\!f||_{L^{\!2}(\widehat{\mathcal {D}}_{\!t\!})}\!$ by the inductive assumption since $\ell \!\le \!m\!-\!1$.

We now consider the case that at least one of $k_1,\cdots , k_s\ge r-2$ so that $\ell \le 2$. Since $r\ge 5$, at most one of the $k_j$, say $k_1$, can be greater than or equal to $r-2$. If $k_1=r-2$ or $k_1=r-1$, then by Sobolev embedding we control $||{\mathcal {T}^{}}^{k_1}\partial \widehat{x}||_{L^{3}(\Omega ^{d_0})} \le C ||{\mathcal {T}^{}}\widehat{x}||_{H^{(r-1, 1/2)}(\Omega ^{d_0})}$, and the other terms involving $\widehat{x}$ can be controlled in $L^{\infty }$ and hence by $||\widehat{x}||_{H^{r-1} (\Omega ^{d_0})}$. Using the estimate (C.5), the inductive assumption and Hölder’s inequality $||f_1f_2||_{L^2}\! \le \! ||f_1||_{L^6}||f_2||_{L^3}$, we control the $L^2$ norm of right-hand side of (C.6) by the right-hand side of (C.1).

The only remaining case is when $k_1=r$, and to deal with this we bound ${\mathcal {T}^{}}^{r}\partial \widehat{x}$ in $L^2$ and use (C.4) to bound the $L^\infty$ norm of the term involving f, which gives

$$\begin{aligned}&||\!{{\,\mathrm{div}\,}}\!{\mathcal {T}^{}}^{m\!} \widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t\!)}\\&\quad \!\! \le \! C_r\bigg (\! ||{\mathcal {T}^{}}^{m\!} g||_{L^2(\widehat{\mathcal {D}}_t\!)}\!+\!\!{\sum }_{\ell =0,1,2}||{\mathcal {T}^{}}^\ell \! g||_{L^6(\widehat{\mathcal {D}}_t\!)}\\&\quad \!+\,\!||g||_{{L}^{\!\infty }(\widehat{\mathcal {D}}_t\!)} \!+\!(||{\mathcal {T}^{}}\widehat{x}||_{H^{r\!}(\Omega ^{d_0}\!)} \!+\! 1){\sum }_{k\le m-\!1}||\widehat{\partial }{\mathcal {T}^{}}^k \widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t\!)}\!\bigg ). \end{aligned}$$

By the inductive assumption, $||\!{{\,\mathrm{div}\,}}\!{\mathcal {T}^{}}^m \widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}$ is controlled by the right-hand side of (C.2). A similar argument shows that $||\!{{\,\mathrm{curl}\,}}\! {\mathcal {T}^{}}^m \widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}$ is bounded by the right-hand side of (C.1) (resp. (C.2)) with $s\! =\! m$ but with $||\widetilde{x}||_{H^m(\Omega )}, ||{\mathcal {T}^{}}\widetilde{x}||_{H^m(\Omega )}$ replaced by $||\widehat{x}||_{H^m(\Omega ^{d_0})}, ||{\mathcal {T}^{}}\widehat{x}||_{H^m(\Omega ^{d_0})}$. Using (7.2) completes the proof. $\quad \square$

We also need the following estimate for the Newton potential:

Lemma C.2

If g is a smooth function supported in $\widehat{x}(t, \Omega ^{d_0\!/2})$, then there is a constant C with

$$\begin{aligned} |\widehat{\partial }^s (g * \Phi )(x)| \le C||g||_{L^2(\widehat{\mathcal {D}}_t)},\qquad x\in \partial \widehat{\mathcal {D}}_t,\qquad s\ge 0. \end{aligned}$$

Proof

Since there exists $c_0>0$ such that $d(\widehat{x}(t, \Omega ^{d_0/2}), \partial \widehat{\mathcal {D}}_t)\ge c_0$, we have that $d(x, z)\ge c_0$ for each $z\in \text {supp}(g)\subset \widehat{x}(t, \Omega ^{d_0/2})$, and so $\widehat{\partial }^s \Phi (x-\cdot )\in L^2(\widehat{x}(t, \Omega ^{d_0/2}))$. Therefore,

$$\begin{aligned} |\widehat{\partial }^s (g*\Phi )| \le ||g||_{L^2(\widehat{\mathcal {D}}_t)}||\widehat{\partial }^s\Phi (x-z)||_{L^2_z(\widehat{x}(t, \Omega ^{d_0/2})} \le C||g||_{L^2(\widehat{\mathcal {D}}_t)}. \end{aligned}$$

$\square$

Proof of Theorem 7.2

We proceed by induction. Write $f = g * \Phi$. When $j=0$ we have

$$\begin{aligned} ||\widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}^2=\int _{\widehat{\mathcal {D}}_t}\delta ^{ij}(\widehat{\partial }_i f)\cdot (\widehat{\partial }_j f)\,\mathrm{d}x =\int _{\partial \widehat{\mathcal {D}}_t} N^i (\widehat{\partial }_i f)f\,\mathrm{d}S(x)-\int _{\widehat{\mathcal {D}}_t}gf\,\mathrm{d}x. \end{aligned}$$

(C.7)

By Lemma C.2, the boundary integral in (C.7) is bounded by $C||g||_{L^2(\widehat{\mathcal {D}}_t)}^2$. The second term in (C.7) is bounded by $||g||_{L^2(\widehat{\mathcal {D}}_t)}||f||_{L^2(\widehat{\mathcal {D}}_t)}$, and by Young’s inequality,

$$\begin{aligned} ||f||_{L^2(\widehat{\mathcal {D}}_t)}=||g*\Phi ||_{L^2(\widehat{\mathcal {D}}_t)}\le C||g||_{L^{2}(\widehat{\mathcal {D}}_t)} ||\Phi ||_{L^{1}(\widehat{\mathcal {D}}_t)}\le C||g||_{L^2(\widehat{\mathcal {D}}_t)}, \end{aligned}$$

By (C.7), this implies

$$\begin{aligned} ||\widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)} \le C||g||_{L^2(\widehat{\mathcal {D}}_t)}. \end{aligned}$$

Suppose that we now know that $||{\mathcal {T}^{}}^j \widehat{\partial }g||_{L^2(\widetilde{\mathcal {D}}_t)}$ is bounded by the right-hand side of () for $j= 0,\cdots , m-1 \le r-1$. To prove that it holds for $j = m$ as well, we integrate by parts to get

$$\begin{aligned}&||{\mathcal {T}^{}}^m\widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}^2 = \int _{\widehat{\mathcal {D}}_t}({\mathcal {T}^{}}^m\widehat{\partial }_i f)({\mathcal {T}^{}}^m\widehat{\partial }^i f)\,\mathrm{d}x\nonumber \\&\quad =\!\int _{\widehat{\mathcal {D}}_t}\underbrace{\!\!\!\delta ^{ij} ({\mathcal {T}^{}}^m\widehat{\partial }_i f)\widehat{\partial }_j{\mathcal {T}^{}}^m \!f \mathrm{d}x}_{I} -\!\int _{\widehat{\mathcal {D}}_t}\underbrace{ \!\!\!\delta ^{ij}({\mathcal {T}^{}}^m\widehat{\partial }_i f) (\widehat{\partial }_j{\mathcal {T}^{}}^m \widehat{x})\widehat{\partial }f \mathrm{d}x}_{II}\\&\qquad +\!\sum \! \int _{\widehat{\mathcal {D}}_t}\underbrace{\!\!\!({\mathcal {T}^{}}^m \widehat{\partial }f)(\partial {\mathcal {T}^{}}^{\ell _1}\widehat{x})\cdots (\partial {\mathcal {T}^{}}^{\ell _{s-1}}\widehat{x}) {\mathcal {T}^{}}^{\ell _s}\widehat{\partial }f \mathrm{d}x}_{III}, \end{aligned}$$

where the sum is over $\ell _1+\cdots +\ell _s=m$ and $\ell _1,\cdots , \ell _s\le m-1$, $\ell _1\ge 1$. To control III, we note that if $\ell _1,\cdots , \ell _{s-1}\le r-3$, then $III\le C(||\widetilde{x}||_{H^{r-1}(\Omega )})||{\mathcal {T}^{}}^m \widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)} ||{\mathcal {T}^{}}^{\ell _s}\widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}$, and we control $||{\mathcal {T}^{}}^{\ell _s}\widehat{\partial }f||_{L^2(\mathcal {D}_t)}$ by the inductive assumption. On the other hand, since $r \ge 5$, there can be at most one j with $\ell _j \ge r-2$ and without loss of generality it is $\ell _1$ in which case $\ell _j\le 2$ for $j=2,3,\cdots , s$. We then bound $III\le C(||\widetilde{x}||_{H^{r-1}(\Omega )}) ||{\mathcal {T}^{}}^m \widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}||\partial {\mathcal {T}^{}}^{\ell _1}\widehat{x}||_{L^3(\widehat{\mathcal {D}}_t)} ||{\mathcal {T}^{}}^{\ell _s}\widehat{\partial }f||_{L^6(\widehat{\mathcal {D}}_t)}$. By Sobolev embedding, $||\partial {\mathcal {T}^{}}^{\ell _1}\widehat{x}||_{L^3(\widehat{\mathcal {D}}_t)} \le C||{\mathcal {T}^{}}\widetilde{x}||_{H^{(r-1, 1/2)}(\Omega )}$, and $||{\mathcal {T}^{}}^{\ell _s}\widehat{\partial }f ||_{L^6(\widehat{\mathcal {D}}_t)}$ can be controlled using Lemma C.1.

To control $I+II$, we integrate by parts and get

$$\begin{aligned}&I+II\!=-\!\int _{\widehat{\mathcal {D}}_t}\!\!\!{}_{\!}\delta ^{ij\!}\!\underbrace{(\widehat{\partial }_i {\mathcal {T}^{}}^{m\!}\widehat{\partial }_j f){\mathcal {T}^{}}^m\! f \mathrm{d}x}_{I_1} +\!\int _{\widehat{\mathcal {D}}_t}\!\!\!{}_{\!}\delta ^{ij\!}\!\underbrace{(\widehat{\partial }_i{\mathcal {T}^{}}^{m\!}\widehat{\partial }_j f)({\mathcal {T}^{}}^m \widehat{x}^k)\widehat{\partial }_k f \mathrm{d}x}_{II_1} \nonumber \\&\quad +\!\int _{\widehat{\mathcal {D}}_t}\!\!\!{}_{\!}\delta ^{ij\!}\!\underbrace{({\mathcal {T}^{}}^m\widehat{\partial }_i f)({\mathcal {T}^{}}^m \widehat{x}^k)\widehat{\partial }_j\widehat{\partial }_k f \mathrm{d}x}_{II_2} +{\mathcal {B}}, \end{aligned}$$

(C.8)

where

$$\begin{aligned} {\mathcal {B}}=\int _{\partial \widehat{\mathcal {D}}_t}(N^i {\mathcal {T}^{}}^m\widehat{\partial }_if)\big ({\mathcal {T}^{}}^m f-({\mathcal {T}^{}}^m\widehat{x}^k)(\widehat{\partial }_k f)\big ). \end{aligned}$$

(C.9)

To control $II_1$, we have

$$\begin{aligned} \delta ^{ij}\widehat{\partial }_i{\mathcal {T}^{}}^m\widehat{\partial }_j f= {\mathcal {T}^{}}^m \Delta f + (\partial {\mathcal {T}^{}}^m \widehat{x})(\widehat{\partial }^2 f)+\sum (\partial {\mathcal {T}^{}}^{\ell _1} \widehat{x})\cdots (\partial {\mathcal {T}^{}}^{\ell _{s-1}} \widehat{x})(\widehat{\partial }{\mathcal {T}^{}}^{\ell _s}\widehat{\partial }f), \end{aligned}$$

(C.10)

where the sum is over $\ell _1+\cdots +\ell _s=m$ and $\ell _1,\cdots , \ell _s\le m-1$. The terms in the sum can be controlled similarly to how we controlled the sum in (C.6). The two main terms that are left in $II_1$ are

$$\begin{aligned} \int _{\widehat{\mathcal {D}}_t}({\mathcal {T}^{}}^m g)({\mathcal {T}^{}}^{m} \widehat{x})(\widehat{\partial }f)\,\mathrm{d}x+\int _{\widehat{\mathcal {D}}_t}(\partial {\mathcal {T}^{}}^m \widehat{x}) (\widehat{\partial }^2f)({\mathcal {T}^{}}^m \widehat{x})(\widehat{\partial }f)\,\mathrm{d}x. \end{aligned}$$

(C.11)

To control the second term in (C.11), we commute one ${\mathcal {T}^{}}$ to the outside which gives

$$\begin{aligned} \!\int _{\widehat{\mathcal {D}}_t}\underbrace{ ({\mathcal {T}^{}}\partial {\mathcal {T}^{}}^{m-1} \widehat{x})(\widehat{\partial }^2 f)({\mathcal {T}^{}}^{m} \widehat{x})(\widehat{\partial }f)\,\mathrm{d}x}_{II_{11}} +\!\int _{\widehat{\mathcal {D}}_t}\underbrace{ (\partial ^2 \widehat{x})(\partial {\mathcal {T}^{}}^{m-1} \widehat{x})(\widehat{\partial }^2f)({\mathcal {T}^{}}^{m} \widehat{x}) (\widehat{\partial }f)\,\mathrm{d}x}_{II_{12}}. \end{aligned}$$

(C.12)

To control $II_{11}$, we integrate half a tangential derivative by parts using (A.3) and get

$$\begin{aligned} II_{11} \le C||\partial {\mathcal {T}^{}}^{m-1} \widehat{x}||_{H^{(0,1/2)}(\Omega )} ||(\widehat{\partial }^2 f)({\mathcal {T}^{}}^m \widehat{x})(\widehat{\partial }f)||_{H^{(0,1/2)}(\widehat{\mathcal {D}}_t)}. \end{aligned}$$

Using the fractional product rule (A.4), for each $\mu$ we have with $L^2=L^2(\widehat{\mathcal {D}}_t)$

$$\begin{aligned}&||\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu \big ((\widehat{\partial }^2\! f)({\mathcal {T}^{}}^m \widehat{x})\widehat{\partial }f\big )||_{L^2} \!\le C ||(\widehat{\partial }^2 \! f)(\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu {\mathcal {T}^{}}^m \widehat{x})\widehat{\partial }f||_{L^2}\! \nonumber \\&\quad +||{\mathcal {T}^{}}^m \widehat{x}||_{L^2(\Omega )} {\sum }_{\ell \le 2}||{\mathcal {T}^{}}^{\ell \!}\big ((\widehat{\partial }^2\! f) \widehat{\partial }f\big )||_{L^2}. \end{aligned}$$

(C.13)

The first term on the right hand side can be controlled by $||{\mathcal {T}^{}}\widetilde{x}||_{H^{(m-1, 1/2)}(\Omega )}||\widehat{\partial }^2 f||_{L^\infty (\widehat{\mathcal {D}}_t)} ||\widehat{\partial }f||_{L^\infty (\widehat{\mathcal {D}}_t)}$. Using (C.4), the Sobolev inequality $||\widehat{\partial }f||_{L^\infty (\widehat{\mathcal {D}}_t)}\le C{\sum }_{|I|\le 1}||\widehat{\partial }^{I+1} f||_{L^6(\widehat{\mathcal {D}}_t)}$ and (C.3), we control this term. To control the second term in (C.13), we just show how to control $||({\mathcal {T}^{}}^\ell \widehat{\partial }^2 f)(\widehat{\partial }f)||_{L^2(\widehat{\mathcal {D}}_t)}$ for $\ell \le 2$ since the remaining terms are similar. For $\ell \le 2$ we have

$$\begin{aligned}&||{\mathcal {T}^{}}^\ell \widehat{\partial }^2 f||_{L^6(\widehat{\mathcal {D}}_t)} \le ||\widehat{\partial }{\mathcal {T}^{}}^\ell \widehat{\partial }f||_{L^6(\widehat{\mathcal {D}}_t)}\\&\quad +{\sum }_{\ell \le 2} {\sum }_{j_1+j_2=\ell , j_1\ge 1}||({\mathcal {T}^{}}^{j_1} \widehat{A})(\widehat{\partial }{\mathcal {T}^{}}^{j_2}\widehat{\partial }f)||_{L^6(\widehat{\mathcal {D}}_t)}. \end{aligned}$$

By (C.3) we control the first term here, and after bounding the term involving $\widehat{A}$ in $L^\infty$ and using (C.3) again we also control the second term. To control the term $II_{12}$ from (C.12), we have

$$\begin{aligned} II_{12}\le P(||{\mathcal {T}^{}}\widetilde{x}||_{H^{r-1}(\Omega )})||\widehat{\partial }^2f||_{L^\infty (\widehat{\mathcal {D}}_t)}||\widehat{\partial }f||_{L^\infty (\widehat{\mathcal {D}}_t)}, \end{aligned}$$

and then use (C.4). To control the first term in (C.11) we use (A.3) and then bound

$$\begin{aligned}&||\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu \big (({\mathcal {T}^{}}^m\widehat{x})(\widehat{\partial }f)\big )||_{L^2} \le C \big (||(\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu {\mathcal {T}^{}}^m\widehat{x})(\widehat{\partial }f)||_{L^2}\\&\quad \!+\! ||\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu \big (({\mathcal {T}^{}}^m\widehat{x})(\widehat{\partial }f)\big )\!-\!(\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu {\mathcal {T}^{}}^m\widehat{x})(\widehat{\partial }f)||_{L^2}\big ), \end{aligned}$$

where $L^2=L^2(\widehat{\mathcal {D}}_t)$, and then

$$\begin{aligned}&||(\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu {\mathcal {T}^{}}^m\widehat{x})(\widehat{\partial }f)||_{L^2(\widehat{\mathcal {D}}_t)}\le ||{\mathcal {T}^{}}\widetilde{x}||_{H^{(r-1, 1/2)}(\Omega )} ||\widehat{\partial }f||_{L^\infty (\widehat{\mathcal {D}}_t)}\\&\quad \le ||{\mathcal {T}^{}}\widetilde{x}||_{H^{(r-1, 1/2)}(\Omega )} {\sum }_{\ell \le 1}||\widehat{\partial }^{\ell +1}f||_{L^6(\widehat{\mathcal {D}}_t)}, \end{aligned}$$

and by (A.4),

$$\begin{aligned}&||\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu \big (({\mathcal {T}^{}}^m\widehat{x})(\widehat{\partial }f)\big )-(\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu {\mathcal {T}^{}}^mx)(\widehat{\partial }f)||_{L^2(\widehat{\mathcal {D}}_t)} \\&\quad \le C||{\mathcal {T}^{}}^m \widetilde{x}||_{L^2(\Omega )}{\sum }_{\ell \le 2}||{\mathcal {T}^{}}^{\ell }\widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}. \end{aligned}$$

To control $II_2$ in (C.8), we have

$$\begin{aligned} II_2 \le ||{\mathcal {T}^{}}^m \widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}||{\mathcal {T}^{}}^m \widehat{x}||_{L^3(\mathcal {D}_t)}||\widehat{\partial }^2 f||_{L^6(\mathcal {D}_t)}, \end{aligned}$$

and $||\widehat{\partial }^2 f||_{L^6(\widehat{\mathcal {D}}_t)}$ under control, using (C.3).

To control $I_1$, we substitute (C.10) into $I_1$ and get, to highest order,

$$\begin{aligned} \int _{\widehat{\mathcal {D}}_t} ({\mathcal {T}^{}}^{m} g)({\mathcal {T}^{}}^{m} f)+\int _{\widehat{\mathcal {D}}_t}({\mathcal {T}^{}}\partial {\mathcal {T}^{}}^{m-1} \widehat{x})(\widehat{\partial }^2 f)({\mathcal {T}^{}}^{m} f). \end{aligned}$$

(C.14)

We write ${\mathcal {T}^{}}= {\mathcal {T}^{}}^a \partial _{y^a}={\mathcal {T}^{}}^a \widehat{A}_a^i \widehat{\partial }_i$, so that

$$\begin{aligned}&{\mathcal {T}^{}}^m \!f\!= ({\mathcal {T}^{}}^a \widehat{A}_a^i \widehat{\partial }_i){\mathcal {T}^{}}^{m\!-\!1}\!f\!={\mathcal {T}^{}}^a \! \widehat{A}_a^i\Big ({\mathcal {T}^{}}^{m\!-\!1}\widehat{\partial }_i f+ (\widehat{\partial }_i{\mathcal {T}^{}}^{m\!-\!1}\widehat{x})(\widehat{\partial }f)\\&\quad +{\sum }_{\ell _1\!+\cdots +\ell _s\le m\!-\!2} (\partial {\mathcal {T}^{}}^{\ell _1}\widehat{x})\cdots (\partial {\mathcal {T}^{}}^{\ell _{s-1}}\widehat{x}){\mathcal {T}^{}}^{\ell _s}\widehat{\partial }f\Big ). \end{aligned}$$

Substituting this into (C.14), to highest order the result is

$$\begin{aligned}&\int _{\widehat{\mathcal {D}}_t}({\mathcal {T}^{}}^m g)({\mathcal {T}^{}}^a\widehat{A}_a^i)({\mathcal {T}^{}}^{m-1}\widehat{\partial }_i f)+ \int _{\widehat{\mathcal {D}}_t}({\mathcal {T}^{}}^m g)({\mathcal {T}^{}}^a\widehat{A}_a^i)(\widehat{\partial }_i {\mathcal {T}^{}}^{m-1} \widehat{x})(\widehat{\partial }f)\\&\quad +\int _{\widehat{\mathcal {D}}_t}({\mathcal {T}^{}}\partial {\mathcal {T}^{}}^{m-1} \widehat{x})(\widehat{\partial }^2 f)({\mathcal {T}^{}}^a \widehat{A}^i_a)({\mathcal {T}^{}}^{m-1}\widehat{\partial }_i f) \\&\quad +\int _{\widehat{\mathcal {D}}_t}({\mathcal {T}^{}}\partial {\mathcal {T}^{}}^{m-1} \widehat{x})(\widehat{\partial }^2 f)({\mathcal {T}^{}}^a \widehat{A}^i_a)(\widehat{\partial }_i {\mathcal {T}^{}}^{m-1} \widehat{x})(\widehat{\partial }f). \end{aligned}$$

The first and third terms can be controlled after integrating ${\mathcal {T}^{}}$ by parts and using Hölder’s inequality. The other terms can be controlled after integrating half a tangential derivative by parts using (A.3) and (A.4).

Finally, to control the boundary term ${\mathcal {B}}$ in (C.9), we use Lemma C.2 to get

$$\begin{aligned} {\mathcal {B}} = \int _{\partial \widehat{\mathcal {D}}_t}\!\!\!(N^i {\mathcal {T}^{}}^r\widehat{\partial }_if)\Big ({\mathcal {T}^{}}^r f-({\mathcal {T}^{}}^r\widehat{x})(\widehat{\partial }f)\Big ) \le C\Big (||g||_{L^2(\widehat{\mathcal {D}}_t)}^2+ ||g||_{L^2(\widehat{\mathcal {D}}_t)}^2\int _{\partial \widehat{\mathcal {D}}_t}\!\!\!|{\mathcal {T}^{}}^{r}\widehat{x}|\Big ), \end{aligned}$$

which is controlled by $C(||{\mathcal {T}^{}}\widetilde{x}||_{H^{(r-1, 0.5)}(\Omega )} + 1)||g||_{L^2(\widehat{\mathcal {D}}_t)}$ by the trace lemma (A.21) and Theorem A.7. $\quad \square$

1.2 C.2: Estimates for Section 7.2

Let ${\mathfrak {D}}^r$ be the mixed tangential space and time derivative defined in Section 3.3. We have

Lemma C.3

Suppose that $r\ge 5$. If $\widehat{\Delta }f=g$ in $\widehat{\mathcal {D}}_t$, then for $j\le r-1$,

$$\begin{aligned}&||\widehat{\partial }{\mathfrak {D}}^j \widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}\le C^\prime _r \bigg ({\sum }_{k=0}^{j+1}||{\mathfrak {D}}^k\widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}\!\\&\quad +{\sum }_{k\le 2}||{\mathfrak {D}}^k g||_{L^6(\widehat{\mathcal {D}}_t)}+||{\mathfrak {D}}^j\! g||_{L^2(\widehat{\mathcal {D}}_t)}+||g||_{L^\infty (\widehat{\mathcal {D}}_t)}\bigg ). \end{aligned}$$

In addition, we have

$$\begin{aligned}&||\widehat{\partial }{\mathfrak {D}}^r \widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}\le P\big (||\widetilde{x}||_{H^{r}(\Omega )}, {{{\,\mathrm{{\textstyle {\sum }}}\,}}}_{k\le r-1}||D_t^kS_\varepsilon V||_{H^{r-k}(\Omega )}\big )\\&\quad \cdot \bigg ({\sum }_{k=0}^{r+1}||{\mathfrak {D}}^k\widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t\!)}\!+||{\mathfrak {D}}^r\! g||_{L^2(\widehat{\mathcal {D}}_t\!)}+||{\mathcal {T}^{}}\widetilde{x}||_{H^{r\!}(\Omega )}\big [{\sum }_{k\le 4}||{\mathfrak {D}}^k \widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t\!)}\\&\quad +\!{\sum }_{k\le 2}||{\mathfrak {D}}^k \! g||_{L^6(\widehat{\mathcal {D}}_t\!)}+||g||_{L^\infty (\widehat{\mathcal {D}}_t\!)}\big ]\bigg ). \end{aligned}$$

Moreover, for $0\le \ell \le 2$,

$$\begin{aligned}&||\widehat{\partial }{\mathfrak {D}}^\ell \widehat{\partial }f||_{L^6(\widehat{\mathcal {D}}_t)} \le C_r^\prime \bigg ({\sum }_{k\le \ell }||{\mathfrak {D}}^k g||_{L^6(\widehat{\mathcal {D}}_t)}\nonumber \\&\quad +{\sum }_{k\le \ell +2}||{\mathfrak {D}}^k \widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}\bigg ). \end{aligned}$$

(C.15)

Proof

It suffices to prove

$$\begin{aligned}&||\widehat{\partial }{\mathfrak {D}}^{r-1}D_t \widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}\nonumber \\&\quad \le C\big (||\widehat{x}||_{H^{r}(\Omega ^{d_0})}, {{{\,\mathrm{{\textstyle {\sum }}}\,}}}_{k\le r-1}||D_t^k\widehat{V}||_{H^{r-k}(\Omega ^{d_0})}\big )\nonumber \\&\quad \bigg ({\sum }_{k=0}^{r+1}||{\mathfrak {D}}^k\widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}\!+||{\mathfrak {D}}^r\! g||_{L^2(\widehat{\mathcal {D}}_t)}\nonumber \\&\quad +||{\mathcal {T}^{}}\widetilde{x}||_{H^{(r-1, 0.5)}(\Omega ^{d_0})}\big [{\sum }_{k\le 4}||{\mathfrak {D}}^k \widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}\nonumber \\&\quad +{\sum }_{k\le 2}||{\mathfrak {D}}^k g||_{L^6(\widehat{\mathcal {D}}_t)}+||g||_{L^\infty (\widehat{\mathcal {D}}_t)}\big ]\bigg ), \end{aligned}$$

(C.16)

because (C.15) will then follow from this estimate and Lemma C.1. Suppose that (C.16) is known for $||\widehat{\partial }{\mathfrak {D}}^{r-1}D_t \widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}$ with $j=1,\cdots , r-2$, then for $j=r-1$, we have

$$\begin{aligned}&||\widehat{\partial }{\mathfrak {D}}^{r-1}D_t \widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}\le ||{{\,\mathrm{div}\,}}{\mathfrak {D}}^{r-1} D_t \widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}\\&\quad + ||{{\,\mathrm{curl}\,}}{\mathfrak {D}}^{r-1}D_t \widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}+ ||{\mathcal {T}^{}}{\mathfrak {D}}^{r-1}D_t\widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}. \end{aligned}$$

Here ${{\,\mathrm{div}\,}}$ and ${{\,\mathrm{curl}\,}}$ stand for the divergence and curl with respect to $\widehat{\partial }$. We only need to control the div term, because the curl term can be treated similarly. Since ${{\,\mathrm{div}\,}}{\mathfrak {D}}^{r-1} D_t \widehat{\partial }_{\!} f\!=\!{\mathfrak {D}}^{r-1}D_t g+\sum ({\mathfrak {D}}^k \widehat{A})(\widehat{\partial }{\mathfrak {D}}^\ell \widehat{\partial }_{\!} f)$, where $\widehat{A}=(\widehat{A}_{\,\,i}^a)$ and the sum is over $k+\ell =r$ such that $k\ge 1$, we have

$$\begin{aligned} {{\,\mathrm{div}\,}}{\mathfrak {D}}^{r-1} D_t \widehat{\partial }f = {\mathfrak {D}}^{r-1} D_t g+\sum ({\mathfrak {D}}^{k_1}\partial \widehat{x})\cdots ({\mathfrak {D}}^{k_s}\partial \widehat{x})(\widehat{\partial }{\mathfrak {D}}^\ell \widehat{\partial }f). \end{aligned}$$

(C.17)

The above sum is over $k_1+\cdots +k_s+\ell =k+\ell =r$, which needs to be controlled in $L^2(\widehat{\mathcal {D}}_t)$. If $\ell \ge 3$, then $k_1,\cdots , k_s\le r-3$, and so all terms involving $\widehat{x}$ can then be controlled in $L^\infty$ by either $||\widehat{x}||_{H^r(\Omega ^{d_0})}$ or ${\sum }_{k\le r-3}||D_t^k \widehat{V}||_{H^{r-k}(\Omega ^{d_0})}$. Furthermore, when at least one of $k_1,\cdots , k_s\ge r-2$, since $r\ge 5$, there is at most one term, say $k_1$, can be greater than or equal to $r-2$. If $k_1=r-2$ or $k_1=r-1$, we control $||{\mathfrak {D}}^{k_1}\partial \widehat{x}||_{L^{3}(\Omega ^{d_0})}$ by either $||{\mathcal {T}^{}}\widehat{x}||_{H^{(r-1, 0.5)}(\Omega ^{d_0})}$ or ${\sum }_{k\le r-2}||D_t^k \widehat{V}||_{H^{r-k}(\Omega ^{d_0})}$, and other terms involving $\widehat{x}$ are of lower order. In addition to this, we control $\widehat{\partial }{\mathfrak {D}}^\ell \widehat{\partial }f$ for $\ell \le 2$ in $L^6$ because by the pointwise inequality (5.5) we have

$$\begin{aligned}&||\widehat{\partial }{\mathfrak {D}}^\ell \widehat{\partial }f||_{L^6(\widehat{\mathcal {D}}_t)} \le C(M)\big ( ||{{\,\mathrm{div}\,}}{\mathfrak {D}}^\ell \widehat{\partial }f||_{L^6(\widehat{\mathcal {D}}_t)}+ ||{{\,\mathrm{curl}\,}}{\mathfrak {D}}^\ell \widehat{\partial }f||_{L^6(\widehat{\mathcal {D}}_t)}\\&\quad +| |{\mathcal {T}^{}}{\mathfrak {D}}^\ell \widehat{\partial }f||_{L^6(\widehat{\mathcal {D}}_t)}\big )\\&\quad \le C(M)\bigg (||{\mathfrak {D}}^\ell g||_{L^6(\widehat{\mathcal {D}}_t)} \\&\quad +{\sum }_{\ell _1+\ell _2=\ell , \ell _1\ge 1}||({\mathfrak {D}}^{\ell _1} \widehat{A})(\widehat{\partial }{\mathfrak {D}}^{\ell _2}\widehat{\partial }f)||_{L^6(\widehat{\mathcal {D}}_t)}+||{\mathcal {T}^{}}{\mathfrak {D}}^\ell \widehat{\partial }f||_{H^1(\widehat{\mathcal {D}}_t)}\bigg ), \end{aligned}$$

where the second term is not present if $\ell = 0$. The second and third terms can be bounded by the right-hand side of (C.15) by the inductive assumption. On the other hand, when $k_1\!=\!r$, ${\mathfrak {D}}^{k_1}\!$ involves at least one $D_t$, and so we control ${\mathfrak {D}}^{k_1}\!\partial \widehat{x}$ in $L^2$ by ${\sum }_{k\le r-\!1} ||D_t^k \widehat{V}||_{H^{r-k}(\Omega ^{d_0})}$. We also control $\widehat{\partial }^2 \!f$ in $L^{\!\infty }\!\!$, as in Lemma C.1. $\quad \square$

Lemma C.4

Fix $r \ge 7$. If g is a smooth function such that $\text {supp}(g)\subset \widehat{x}(t, \Omega ^{d_0/2})$, then

$$\begin{aligned}&||D_{\!t}^r \!(g * \Phi )||_{L^{\!2}(\widehat{\mathcal {D}}_t)} \le C_r' \bigg ({\sum }_{k\le r\!-1}||{\mathfrak {D}}^{k} \widehat{\partial }f||_{L^{\!2}(\widehat{\mathcal {D}}_t)}\\&\quad +{\sum }_{k\le r}||{\mathfrak {D}}^k \! g||_{L^{\!2}(\widehat{\mathcal {D}}_t)}+{\sum }_{k\le 2}||{\mathfrak {D}}^k \! g||_{L^6(\widehat{\mathcal {D}}_t)}+||g||_{L^\infty (\widehat{\mathcal {D}}_t)}\!\bigg ). \end{aligned}$$

Proof

Since $\widehat{\Delta } f = g$ in $\widehat{\mathcal {D}}_t$, commuting $D_t^r$ through this and get

$$\begin{aligned} \widehat{\Delta }D_t^r f = (D_t^rg)+ [\widehat{\Delta }, D_t^r] f. \end{aligned}$$

(C.18)

In addition, since $D_t=\partial _t+\widehat{V}^k\widehat{\partial }_k$ in $\widehat{\mathcal {D}}_t$, we have $[\widehat{\partial }, D_t]=\widehat{\partial }\widehat{V}\cdot \widehat{\partial }$, which can then be used to compute

$$\begin{aligned}{}[\widehat{\Delta }, D_t^r]= & {} {\sum }_{\ell _1+\ell _2=r-1}c_{\ell _1,\ell _2} (\widehat{\Delta }D_t^{\ell _1} \widehat{V})\cdot \widehat{\partial }D_t^{\ell _2}+ {\sum }_{\ell _1+\ell _2=r-1}c_{\ell _1,\ell _2} (\widehat{\partial }D_t^{\ell _1} \widehat{V})\cdot \widehat{\partial }D_t^{\ell _2}\widehat{\partial }\nonumber \\&+ {\sum }_{\ell _1+\cdots +\ell _n= r-n+1, \, n\ge 3} d_{\ell _1,\cdots ,\ell _n}(\widehat{\partial }D_t^{\ell _3} \widehat{V})\cdots (\widehat{\partial }D_t^{\ell _{n}} \widehat{V})\cdot (\widehat{\partial }^2 D_t^{\ell _1} \widehat{V})\cdot D_t^{\ell _2}\widehat{\partial }\nonumber \\&+{\sum }_{\ell _1+\cdots + \ell _n= r-n+1, \, n\ge 3} e_{\ell _1,\cdots ,\ell _n}(\widehat{\partial }D_t^{\ell _3} \widehat{V})\cdots (\widehat{\partial }D_t^{\ell _n} \widehat{V})\cdot (\widehat{\partial }D_t^{\ell _1} \widehat{V})\cdot \widehat{\partial }D_t^{\ell _2}\widehat{\partial }. \nonumber \\ \end{aligned}$$

(C.19)

Since $\widehat{x}(t,y)\!=\!x_0(y)$ in $\Omega ^{d_0}\!\setminus \!\Omega ^{d_0/2}$, $[\widehat{\Delta }, D_t^r]f$ is compactly supported in $\widehat{x}(t,\Omega ^{d_0/2})$. Therefore, (C.18) yields

$$\begin{aligned} D_t^r f = (D_t^rg)*\Phi + ([\widehat{\Delta }, D_t^r] f)*\Phi . \end{aligned}$$

(C.20)

The first term on the right can be controlled by $C(\text {Vol}(\widehat{\mathcal {D}}_t))||D_t^r g||_{L^2(\widehat{\mathcal {D}}_t)}$ using Young’s inequality. In addition, by (C.19), to control the $L^2(\widehat{\mathcal {D}}_t)$ norm of the second term it suffices to consider

$$\begin{aligned}&||[(\widehat{\partial }^2 \!D_t^{\ell _{\!1\!}} \widehat{V})_{\!}\cdots _{\!} (\widehat{\partial }D_t^{\ell _{\!n\!-\!1\!}} \widehat{V})\!\cdot \!D_t^{\ell _{\!n\!}}\widehat{\partial }f]\!*\!\Phi ||_{L^{\!2}(\widehat{\mathcal {D}}_t\!)}\nonumber \\&\quad \text {and}\qquad ||[(\widehat{\partial }D_t^{\ell _{\!1\!}} \widehat{V})_{\!}\cdots _{\!} (\widehat{\partial }D_t^{\ell _{\!n\!-\!1\!}} \widehat{V})\!\cdot \!\widehat{\partial }D_t^{\ell _{\!n\!}}\widehat{\partial }f]\!*\!\Phi ||_{L^{\!2}(\widehat{\mathcal {D}}_t\!)}, \end{aligned}$$

(C.21)

where $\ell _1+\cdots +\ell _n=r+1-n$ and $n\ge 2$. For the first term in (C.21), when $\ell _n\ge 3$, we must have $\ell _j\le r-4$ for $j\le n-1$. In this case, we bound the $\widehat{V}$ terms in $L^\infty (\widehat{\mathcal {D}}_t)$ and then use the Sobolev lemma to get

$$\begin{aligned}&||[(\widehat{\partial }^2 \!D_t^{\ell _{\!1\!}} \widehat{V})_{\!}\cdots _{\!} (\widehat{\partial }D_t^{\ell _{\!n\!-\!1\!}} \widehat{V})\!\cdot \!D_t^{\ell _{\!n\!}}\widehat{\partial }f]\!*\!\Phi ||_{L^{\!2}(\widehat{\mathcal {D}}_t\!)} \\&\quad \le C ||(\widehat{\partial }^2 \!D_t^{\ell _{\!1\!}} \widehat{V})_{\!}\cdots _{\!} (\widehat{\partial }D_t^{\ell _{\!n\!-\!1\!}} \widehat{V})\!\cdot \!D_t^{\ell _{\!n\!}}\widehat{\partial }f||_{L^{\!2}(\widehat{\mathcal {D}}_t\!)} \le C'_r|| D_t^{\ell _{\!n\!}}\widehat{\partial }f||_{L^{\!2}(\widehat{\mathcal {D}}_t\!)}. \end{aligned}$$

When $\ell _n=1,2$, the worst case scenario is when $n=2$ and $D_t^{r-1-\ell _n}$ falls on $\widehat{\partial }^2 \widehat{V}$. In other words, we only need to control $||[(\widehat{\partial }^2 D_t^{r-1-\ell _n} \widehat{V})(D_t^{\ell _n} \widehat{\partial }f)]*\Phi ||_{L^2(\widehat{\mathcal {D}}_t)}$. Writing

$$\begin{aligned}&{[}(\widehat{\partial }^2 D_t^{r-1-\ell _n} \widehat{V})(D_t^{\ell _n} \widehat{\partial }f)]*\Phi \\&\quad = \widehat{\partial }[(\widehat{\partial }D_t^{r-1-\ell _n} \widehat{V})(D_t^{\ell _n} \widehat{\partial }f)]*\Phi -(\widehat{\partial }D_t^{r-1-\ell _n} \widehat{V})(\widehat{\partial }D_t^{\ell _n} \widehat{\partial }f)*\Phi \\&\quad =[(\widehat{\partial }D_t^{r-1-\ell _n} \widehat{V})(D_t^{\ell _n} \widehat{\partial }f)]*(\widehat{\partial }\Phi )-(\widehat{\partial }D_t^{r-1-\ell _n} \widehat{V})(\widehat{\partial }D_t^{\ell _n} \widehat{\partial }f)*\Phi , \end{aligned}$$

and using that $\widehat{\partial }\Phi$ and $\Phi$ belong to $L^1(\widehat{\mathcal {D}}_t)$, Young’s inequality implies that

$$\begin{aligned}&||[(\widehat{\partial }D_t^{r\!-\!1\!-\!\ell _{\!n\!}} \widehat{V})D_t^{\ell _{\!n\!}} \widehat{\partial }{}_{\!} f]*\widehat{\partial }\Phi ||_{L^{\!2}(\widehat{\mathcal {D}}_t\!)}\!+||[(\widehat{\partial }D_t^{r\!-\!1\!-\!\ell _{\!n\!}} \widehat{V})\widehat{\partial }D_t^{\ell _{\!n\!}} \widehat{\partial }{}_{\!}f]*\Phi ||_{L^{\!2}(\widehat{\mathcal {D}}_t\!)}\\&\quad \!\lesssim \! {\sum }_{k\le 1}||(\widehat{\partial }D_t^{r\!-\!1\!-\!\ell _{\!n\!}} \widehat{V})\widehat{\partial }^k \! D_t^{\ell _{\!n\!}}\widehat{\partial }{}_{\!} f||_{L^{\!2}(\widehat{\mathcal {D}}_t\!)}. \end{aligned}$$

Next, to control the term on the right hand side, we have

$$\begin{aligned}&{\sum }_{k\le 1}\!||(\widehat{\partial }D_t^{r\!-\!1\!-\!\ell _{\!n\!}} \widehat{V})\widehat{\partial }^k \! D_t^{\ell _{\!n\!}}\widehat{\partial }{}_{\!} f||_{L^{\!2}(_{\!}\widehat{\mathcal {D}}_t\!)}\! \\&\quad \le \! C_r'{\sum }_{k\le 1}\!||\widehat{\partial }D_t^{\ell _{\!n\!}}\widehat{\partial }{}_{\!} f||_{L^{\!6}(_{\!}\widehat{\mathcal {D}}_t\!)}\! \le \! C_{\!r\!}'\big (||\widehat{\partial }D_t^{\ell _{\!n\!}}\widehat{\partial }{}_{\!} f||_{L^{\!6}(_{\!}\widehat{\mathcal {D}}_t\!)} \\&\quad \!+\!{\sum }_{k\le 1}\!||\widehat{\partial }^k \! D_t^{\ell _{\!n\!}}\widehat{\partial }{}_{\!} f||_{L^{\!6}(_{\!}\widehat{\mathcal {D}}_t\!)}\big ), \end{aligned}$$

which can be controlled using Lemma C.3. When $\ell _n=0$, the worst-case scenario is when $n=2$ and $D_t^{r-1}$ falls on $\widehat{\partial }^2 \widehat{V}$. In other words, we only need to control $||(\widehat{\partial }^2 D_t^{r-1} \widehat{V})(\widehat{\partial }f)||_{L^2(\widehat{\mathcal {D}}_t)}$. By a similar argument as above, we need to control ${\sum }_{k=1,2}||(\widehat{\partial }D_t^{r-1} \widehat{V})(\widehat{\partial }^k f)||_{L^2(\widehat{\mathcal {D}}_t)}$, and this requires the control of $||\widehat{\partial }^k f||_{L^\infty (\widehat{\mathcal {D}}_t)}$ for $k=1,2$. The case when $k=2$ is treated in Lemma C.1, and when $k=1$, we have, by Young’s inequality,

$$\begin{aligned} ||\widehat{\partial }f||_{L^\infty (\widehat{\mathcal {D}}_t)} \le ||g*(\widehat{\partial }\Phi )||_{L^\infty (\widehat{\mathcal {D}}_t)} \le C||g||_{L^\infty (\widehat{\mathcal {D}}_t)}. \end{aligned}$$

To control the $L^2(\widehat{\mathcal {D}}_t)$ norm for the second product in (C.21), when $\ell _n=r-1$ and $n=2$, we write

$$\begin{aligned}{}[(\widehat{\partial }\widehat{V})(\widehat{\partial }D_t^{r-1}\widehat{\partial }f)]*\Phi =[(\widehat{\partial }\widehat{V})(D_t^{r-1}\widehat{\partial }f)]*(\widehat{\partial }\Phi )-[(\widehat{\partial }^2 \widehat{V})(D_t^{r-1}\widehat{\partial }f)]*\Phi , \end{aligned}$$

whose $L^2(\widehat{\mathcal {D}}_t)$ norm can then be controlled by $C_r'{\sum }_{k\le r-1}||D_t^k \widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}$ using Young’s inequality and Sobolev’s lemma. When $r-2\ge \ell _n\ge 2$ (and so $\ell _j\le r-3$ for $j=1,\cdots , n-1)$, we have

$$\begin{aligned} ||[(\widehat{\partial }D_t^{\ell _1} \widehat{V})\cdots (\widehat{\partial }D_t^{\ell _{n-1}} \widehat{V})\cdot (\widehat{\partial }D_t^{\ell _n} \widehat{\partial }f)]*\Phi ||_{L^2(\widehat{\mathcal {D}}_t)} \le C'_r||\widehat{\partial }D_t^{\ell _n} \widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}, \end{aligned}$$

using Young’s inequality and Sobolev’s lemma. The right hand side is controlled by Lemma C.3. If $\ell _n\!=\!1$, it suffices to consider $||[(\widehat{\partial }D_t^{r-2} \widehat{V})\widehat{\partial }D_t \widehat{\partial }f]\!*\!\Phi ||_{L^2(\widehat{\mathcal {D}}_{t\!})}$, which is bounded by $C'_r||\widehat{\partial }D_t\widehat{\partial }f||_{L^6(\widehat{\mathcal {D}}_{t\!})}$. When $\ell _n\!=\!0$, we need to control $||(\widehat{\partial }D_t^{r-1}\widehat{V})\widehat{\partial }^2 \!f]\!*\!\Phi ||_{L^{\!2}(_{\!}\widehat{\mathcal {D}}_{t\!})}$, which requires control of $||\widehat{\partial }^2\! f||_{L^{\!\infty }(\widehat{\mathcal {D}}_{t\!})}$ as in in Lemma C.1. $\quad \square$

Lemma C.5

There is a constant $C\!$ so that if g is smooth and supported in $\widehat{x}({}_{\!}t,{}_{\!} \Omega ^{d_0\!/2})$ and $f\!{}_{\!}=\!g{}_{\!}*{}_{\!}\Phi$ then

$$\begin{aligned} |\widehat{\partial }^s D_t^k (g*\Phi )(x)| \le C||D_t^k g||_{L^2(\widehat{\mathcal {D}}_t)},\qquad x\in \partial \widehat{\mathcal {D}}_t\qquad k, s\ge 0. \end{aligned}$$

(C.22)

Proof

We have $[\widehat{\Delta }, D_t^k] f(x)\!=\!0$ when $x\!\in \! \partial \widehat{\mathcal {D}}_t$ since $\widehat{V}\!\!=\!0$ near $\partial \widehat{\mathcal {D}}_t$,. Therefore, (C.20) yields $\widehat{\partial }^s \! D_t^k\! f(x)\!=\!(D_t^k g)\!*\!(\widehat{\partial }^s\Phi )(x)$ and so (C.22) follows from a similar argument as in the proof of Lemma C.2. $\quad \square$

Proof of Theorem 7.6

It suffices to prove that, for $j \le r-1$,

$$\begin{aligned}&||{\mathfrak {D}}^jD_t \widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)} \le C\big ( ||\widetilde{x}||_{H^{r}(\Omega )}, {{{\,\mathrm{{\textstyle {\sum }}}\,}}}_{k\le r-1} ||D_t^k V||_{H^{r-k}(\Omega )}\big )\nonumber \\&\quad \cdot ||{\mathcal {T}^{}}\widetilde{x}||_{H^{(r-1, 1/2)}(\Omega )} \bigg ({\sum }_{k\le r}||{\mathfrak {D}}^kg||_{L^2(\widehat{\mathcal {D}}_t)}\nonumber \\&\quad +{\sum }_{k\le 2} ||{\mathfrak {D}}^kg||_{L^6(\widehat{\mathcal {D}}_t)}+||g||_{L^\infty (\widehat{\mathcal {D}}_t)}\bigg ). \end{aligned}$$

(C.23)

When $j=r-1$, we have

$$\begin{aligned}&||{\mathfrak {D}}^{r-1}D_t\widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}^2 =\int _{\widehat{\mathcal {D}}_t}\underbrace{\delta ^{ij}({\mathfrak {D}}^{r-1} D_t\widehat{\partial }_i f)(\widehat{\partial }_j{\mathfrak {D}}^{r-1} D_tf)\,\mathrm{d}x}_{I} \\&\quad +\int _{\widehat{\mathcal {D}}_t}\underbrace{\delta ^{ij}({\mathfrak {D}}^{r-1}D_t\widehat{\partial }_i f)([{\mathfrak {D}}^{r-1}D_t, \widehat{\partial }_j]f)}_{II}. \end{aligned}$$

We then control II by applying Corollary D.4. To control I, we integrate by parts and get

$$\begin{aligned} -\int _{\widehat{\mathcal {D}}_t}\underbrace{\delta ^{ij}(\widehat{\partial }_i{\mathfrak {D}}^{r-1} D_t\widehat{\partial }_j f)({\mathfrak {D}}^{r-1} D_tf) \,\mathrm{d}x}_{I_1}+\int _{\partial \widehat{\mathcal {D}}_t}\underbrace{(N^i {\mathfrak {D}}^{r-1}D_t\widehat{\partial }_if) ({\mathfrak {D}}^{r-1}D_t f)}_{{\mathcal {B}}}. \end{aligned}$$

The interior term $I_{\!1}$ is equal to $\int _{\widehat{\mathcal {D}}_t}({\mathfrak {D}}^{r\!-\!1}\!D_t g )({\mathfrak {D}}^{r\!-\!1}\!D_t f)$ to highest order. The error terms here are as in (C.17), and the $L^2$ norm of these terms contribute $||{\mathcal {T}^{}}\widetilde{x}||_{H^{(r-1, 1/2)}(\Omega )}$ in (C.23) using (A.3). When ${\mathfrak {D}}^{r-1}=D_t^r$, this term can be controlled by $||D_t^r g||_{L^2(\widehat{\mathcal {D}}_t)}||D_t^r f||_{L^2(\widehat{\mathcal {D}}_t)}$, and then we may bound $||D_t^r f||_{L^2(\widehat{\mathcal {D}}_t)}$ using Lemma C.4. In addition, when ${\mathfrak {D}}^{r-1}={\mathcal {T}^{}}{\mathfrak {D}}^{r-2}$, we control $I_1$ by integrating ${\mathcal {T}^{}}$ by parts, similar to the control of (C.14) in the proof of Theorem 7.2. Finally, we use Lemma C.5 to control ${\mathcal {B}}$. $\quad \square$

1.3 C.3: Estimates for Section 7.3

Theorem C.6

If $r\ge 5$, then for each $\mu = 0,..., N$ and $j\le r-1$,

$$\begin{aligned}&||\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu {\mathcal {T}^{}}^{j\!} \widehat{\partial }(g*\Phi )||_{L^2(\widehat{\mathcal {D}}_t)} \!\le \! P(||\widetilde{x}||_{H^{r_{\!}}(\Omega )}) \bigg (\!||g||_{L^\infty (\widehat{\mathcal {D}}_t)}\!+\!\!{\sum }_{k\le 2}||{\mathcal {T}^{}}^k \!g||_{L^6(\widehat{\mathcal {D}}_t)}\nonumber \\&\quad \!+\!\!{\sum }_{k\le r_{\!}-_{\!}1}||{\mathcal {T}^{}}^k \! g||_{L^2(\widehat{\mathcal {D}}_t)}\!\bigg ). \end{aligned}$$

(C.24)

Proof

Suppose that we know (C.24) holds for $0\le j\le r-2$, when $j=r-1$, we have

$$\begin{aligned}&||\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu {\mathcal {T}^{}}^{r-1}\widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}^2 = \int _{\widehat{\mathcal {D}}_t}(\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu {\mathcal {T}^{}}^{r-1}\widehat{\partial }_i f)(\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu {\mathcal {T}^{}}^{r-1}\widehat{\partial }^i f)\,\mathrm{d}x\\&\quad =\int _{\widehat{\mathcal {D}}_t}\underbrace{(\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu {\mathcal {T}^{}}^{r-1}\widehat{\partial }_i f)\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu (\widehat{\partial }^i{\mathcal {T}^{}}^{r-1}f)\,\mathrm{d}x}_{I} \\&\quad -\int _{\widehat{\mathcal {D}}_t}\underbrace{(\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu {\mathcal {T}^{}}^{r-1}\widehat{\partial }_i f) \langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu \big ((\widehat{\partial }^i{\mathcal {T}^{}}^{r-1} \widehat{x})(\widehat{\partial }f)\big )\,\mathrm{d}x}_{II}\\&\quad +\sum \int _{\widehat{\mathcal {D}}_t}\underbrace{(\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu {\mathcal {T}^{}}^{r-1} \widehat{\partial }f)\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu \big ((\partial {\mathcal {T}^{}}^{\ell _1}\widehat{x})\cdots (\partial {\mathcal {T}^{}}^{\ell _{s-1}}\widehat{x}) ({\mathcal {T}^{}}^{\ell _s}\widehat{\partial }f)\big )\,\mathrm{d}x}_{III}, \end{aligned}$$

where the sum of over $\ell _1+\cdots +\ell _s =r-1$, $\ell _1,\cdots ,\ell _s\le r-2$. Invoking (A.4), one has, with $L^2=L^2(\widehat{\mathcal {D}}_t)$, that

$$\begin{aligned}&II\!\le \!\! \int _{\widehat{\mathcal {D}}_t}\!\!\!(\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu {\mathcal {T}^{}}^{r_{\!}-_{\!}1}\widehat{\partial }f)(\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu \widehat{\partial }{\mathcal {T}^{}}^{r_{\!}-_{\!}1}\! \widehat{x})(\widehat{\partial }_{\!} f)\mathrm{d}x \\&\quad +C||\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu {\mathcal {T}^{}}^{r_{\!}-_{\!}1}\widehat{\partial }_{\!} f||_{L^2}||\widehat{\partial }{\mathcal {T}^{}}^{r_{\!}-_{\!}1}\! f||_{L^2}{\sum }_{k\le 2}||{\mathcal {T}^{}}^k \widehat{\partial }_{\!} f||_{L^2}. \end{aligned}$$

The last term on the right hand side is of the correct form that we control, while the main term is controlled as the corresponding term (i.e., II) in the proof of Theorem 7.2 and a repeated use of (A.4). In addition,

$$\begin{aligned}&I \!\le \!\!\int _{\widehat{\mathcal {D}}_t}\!\!\!(\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu {\mathcal {T}^{}}^{r_{\!}-_{\!}1}\widehat{\partial }_i f)(\widehat{\partial }^i\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu {\mathcal {T}^{}}^{r-1}\!f)\mathrm{d}x \\&\quad +C||\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu {\mathcal {T}^{}}^{r_{\!}-_{\!}1}\widehat{\partial }_{\!} f||_{L^2(\widehat{\mathcal {D}}_t)}||\widehat{\partial }{\mathcal {T}^{}}^{r_{\!}-_{\!}1}\! f||_{L^2(\widehat{\mathcal {D}}_t)}{\sum }_{k\le 2}||{\mathcal {T}^{}}^k \widehat{A}||_{L^2(\widehat{\mathcal {D}}_t)}. \end{aligned}$$

The last term on the right hand side is of the form that we control, while the main term can be controlled similarly to how we controlled the corresponding term (i.e., I) in the proof of Theorem 7.2 after a repeated use of (A.4). Finally, we need to control the $L^2$ norm of $\sum \langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu ((\partial {\mathcal {T}^{}}^{\ell _1}\widehat{x})\cdots (\partial {\mathcal {T}^{}}^{\ell _{s-1}}\widehat{x}) ({\mathcal {T}^{}}^{\ell _s}\widehat{\partial }f))$ in III. When $\ell _s\ge 3$, then $\ell _1,\cdots , \ell _{s-1}\le r-4$, and so we let $\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu$ fall on $\partial {\mathcal {T}^{}}^{\ell _1} \widehat{x}$ by applying (A.4) and then control the terms involving $\widehat{x}$ in $L^{\infty }$. Moreover, if at least one of $\ell _1,\cdots , \ell _{s-1}$, say $\ell _1$, is greater than or equal to $r-3$, we let $\langle \partial _{\theta {}_{\!}} \rangle ^{\!1_{\!}/2}_\mu$ falls on $\partial {\mathcal {T}^{}}^{\ell _1} \widehat{x}$ by applying (A.4) and control this term in $L^3$, and so ${\mathcal {T}^{}}^s \widehat{\partial }f$ is controlled in $L^6$. But this can then be treated using Sobolev embedding and then Lemma C.1. $\square$

1.4 C.4: Estimates for Section 7.4

Lemma C.7

Suppose that $r\!\ge \! 7$ and $f_J$ satisfy $\widehat{\Delta }_J f_J\!=\!g_J$ for $J=I,{}_{\!}I{\!}I$. Then for $j\!\le \! r-1$, we have

$$\begin{aligned}&||\widehat{\partial }_{{}_{\!}I}{\mathfrak {D}}^j \widehat{\partial }_{{}_{\!}I}f_I-\widehat{\partial }_{{}_{\!}I{\!}I}{\mathfrak {D}}^j \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})} \nonumber \\&\quad \le D_r\bigg ({\sum }_{k\le r}||{\mathfrak {D}}^k \widehat{\partial }_{{}_{\!}I}f_I-{\mathfrak {D}}^k\widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})}+||{\mathfrak {D}}^{r-1} (g_I-g_{{}_{\!}I{\!}I})||_{L^2(\Omega ^{d_0})}\nonumber \\&\quad +{\sum }_{k\le 2}||{\mathfrak {D}}^k (g_I-g_{{}_{\!}I{\!}I})||_{L^6(\Omega ^{d_0})} +||g_I-g_{{}_{\!}I{\!}I}||_{L^\infty (\Omega ^{d_0})} \nonumber \\&\quad +\bigg \{|| \,\widetilde{\!x}_{{}_{\!}I}-\,\widetilde{\!x}_{{}_{\!}I\!I}||_{H^r(\Omega )}+{\sum }_{k\le r-2}||D_t^k (V_{{}_{\!}{}_{\!}I}-V_{{}_{\!}I{\!}I})||_{H^{r-k}(\Omega )}\bigg \}\nonumber \\&\quad \cdot \big ({\sum }_{k\le r}||{\mathfrak {D}}^k \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})}+||{\mathfrak {D}}^{r-1}g_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})}\nonumber \\&\quad + {\sum }_{k\le 2}||{\mathfrak {D}}^k g_{{}_{\!}I{\!}I}||_{L^6(\Omega ^{d_0}\!)}+||g_{{}_{\!}I{\!}I}||_{L^\infty (\Omega ^{d_0}\!)}\big )\bigg ). \end{aligned}$$

(C.25)

where $D_r\!=\!D_r\big (||\,\widetilde{\!x}_{{}_{\!}I}\!||_{H^r(\Omega )}, \!||\,\widetilde{\!x}_{{}_{\!}I\!I}\!||_{H^r(\Omega )},\! {\sum }_{k\le r-2}||D_t^kV_{{}_{\!}{}_{\!}I}\!||_{H^{r\!-\!k}(\Omega )},\! {\sum }_{k\le r-2}||D_t^kV_{{}_{\!}I{\!}I}\!||_{H^{r\!-\!k}(\Omega )}\big )$. For $0\!\le \!\ell \!\le \!2$, we have

$$\begin{aligned}&||\widehat{\partial }_{{}_{\!}I}{\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I}f_I-\widehat{\partial }_{{}_{\!}I{\!}I}{\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^6(\Omega ^{d_0})}\\&\quad \le D_r\bigg ({\sum }_{k\le \ell }||{\mathfrak {D}}^k (g_I-g_{{}_{\!}I{\!}I})||_{L^6(\Omega ^{d_0})}+{\sum }_{k\le \ell +2}||{\mathfrak {D}}^k \widehat{\partial }_{{}_{\!}I}f_I-{\mathfrak {D}}^k \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})}\\&\quad +||\,\widetilde{\!x}_{{}_{\!}I}-\,\widetilde{\!x}_{{}_{\!}I\!I}||_{H^r(\Omega )}{\sum }_{k\le \ell }||\widehat{\partial }_{{}_{\!}I{\!}I}{\mathfrak {D}}^k \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^6(\Omega ^{d_0})}\bigg ), \end{aligned}$$

as well as

$$\begin{aligned}&||\widehat{\partial }_{{}_{\!}I}^2 f_I-\widehat{\partial }_{{}_{\!}I{\!}I}^2 f_{{}_{\!}I{\!}I}||_{L^\infty (\Omega ^{d_0})}\lesssim ||g_I-g_{{}_{\!}I{\!}I}||_{L^\infty (\Omega ^{d_0})}\\&\quad +{\sum }_{\ell \le 1}||\widehat{\partial }_{{}_{\!}I}^\ell {\mathcal {T}^{}}\widehat{\partial }_{{}_{\!}I}f_I-\widehat{\partial }_{{}_{\!}I{\!}I}^\ell {\mathcal {T}^{}}\widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^6(\Omega ^{d_0})} \\&\quad +||\,\widetilde{\!x}_{{}_{\!}I}-\,\widetilde{\!x}_{{}_{\!}I\!I}||_{H^r(\Omega )}{\sum }_{\ell \le 1}||\widehat{\partial }_{{}_{\!}I{\!}I}{\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^6(\Omega ^{d_0})}. \end{aligned}$$

Proof

For $j\!=\!0$ (C.25) follows from (B.2). Suppose that (C.25) hold for $j\!\le \!r\!-\!2$. When $j\!=\!r\!-\!1$, we have

$$\begin{aligned}&||\widehat{\partial }_{{}_{\!}I}{\mathfrak {D}}^{r-1\!} \widehat{\partial }_{{}_{\!}I}f_I-\widehat{\partial }_{{}_{\!}I{\!}I}{\mathfrak {D}}^{r-1\!} \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})}\\&\quad \lesssim ||{{\,\mathrm{div}\,}}_I \!{\mathfrak {D}}^{r-1\!} \widehat{\partial }_{{}_{\!}I}f_I-{{\,\mathrm{div}\,}}_{{}_{\!}I{\!}I}\!{\mathfrak {D}}^{r-1\!}\widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})} +||{{\,\mathrm{curl}\,}}_I\! {\mathfrak {D}}^{r-1\!} \widehat{\partial }_{{}_{\!}I}f_I\\&\quad -{{\,\mathrm{curl}\,}}_{{}_{\!}I{\!}I}\!{\mathfrak {D}}^{r-1\!}\widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})}\,\,\,\,\,\\&\quad +||{\mathcal {T}^{}}{\mathfrak {D}}^{r-1\!} \widehat{\partial }_{{}_{\!}I}f_I-{\mathcal {T}^{}}{\mathfrak {D}}^{r-1\!} \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})}\\&\quad +||\widehat{x}_{{}_{\!}I}-\widehat{x}_{{}_{\!}I{\!}I}||_{H^r(\Omega ^{d_0})}||\widehat{\partial }_{{}_{\!}I{\!}I}{\mathfrak {D}}^{r-1} \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})}. \end{aligned}$$

It suffices to control the ${{\,\mathrm{div}\,}}$ term, since the ${{\,\mathrm{curl}\,}}$ term can be controlled similarly. We have

$$\begin{aligned}&{{\,\mathrm{div}\,}}_{\!I} \!{\mathfrak {D}}^{r-1\!} \widehat{\partial }_{{}_{\!}I}{}_{\!} f_I\!-{{\,\mathrm{div}\,}}_{{}_{\!}I{\!}I}\! {\mathfrak {D}}^{r-1\!} \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}\\&\quad = {\mathfrak {D}}^{r-1}\! (g_I\!-g_{{}_{\!}I{\!}I}) +\!\sum \Big (({\mathfrak {D}}^{k_1\!}\partial \widehat{x}_{{}_{\!}I})\cdots ({\mathfrak {D}}^{k_s\!}\partial \widehat{x}_{{}_{\!}I}{}_{\!})\widehat{\partial }_{{}_{\!}I}{\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I}{}_{\!} f_I\!\\&\quad -({\mathfrak {D}}^{k_1\!}\partial \widehat{x}_{{}_{\!}I{\!}I}{}_{\!})\cdots ({\mathfrak {D}}^{k_s\!}\partial \widehat{x}_{{}_{\!}I{\!}I})\widehat{\partial }_{{}_{\!}I{\!}I}{\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}\Big ), \end{aligned}$$

where the sum is over $k_1\!+\!\cdots \!+\!k_s\!+\!\ell \!=\!r\!-\!1$, $k_1\!\ge \! 1$. To control the sum in $L^2(\Omega ^{d_0}\!)$ we only need to consider

$$\begin{aligned} {\mathcal {A}}&=({\mathfrak {D}}^{k_1}\partial \widehat{x}_{{}_{\!}I})\cdots ({\mathfrak {D}}^{k_s}\partial \widehat{x}_{{}_{\!}I})(\widehat{\partial }_{{}_{\!}I}{\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I}f_I-\widehat{\partial }_{{}_{\!}I{\!}I}{\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}),\\ {\mathcal {B}}&=({\mathfrak {D}}^{k_1}\partial \widehat{x}_{{}_{\!}I}-{\mathfrak {D}}^{k_1}\partial \widehat{x}_{{}_{\!}I{\!}I})({\mathfrak {D}}^{k_2}\partial \widehat{x}_{{}_{\!}I{\!}I})\cdots ({\mathfrak {D}}^{k_s}\partial \widehat{x}_{{}_{\!}I{\!}I})(\widehat{\partial }_{{}_{\!}I{\!}I}{\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}). \end{aligned}$$

Now, if $\ell \ge 2$, then $k_1,\cdots , k_s\le r-3$, and so all terms involving $\widehat{x}$ can then be controlled in $L^\infty$, i.e.,

$$\begin{aligned} ||{\mathcal {A}}||_{L^2(\Omega ^{d_0})}&\le D_r||\widehat{\partial }_{{}_{\!}I}{\mathfrak {D}}^{\ell } \widehat{\partial }_{{}_{\!}I}f_I-\widehat{\partial }_{{}_{\!}I{\!}I}{\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})},\\ ||{\mathcal {B}}||_{L^2(\Omega ^{d_0})}&\le D_r||{\mathfrak {D}}^{k_1}\partial \widehat{x}_{{}_{\!}I}-{\mathfrak {D}}^{k_1}\partial \widehat{x}_{{}_{\!}I{\!}I}||_{L^\infty (\Omega ^{d_0})}||\widehat{\partial }_{{}_{\!}I{\!}I}{\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})}. \end{aligned}$$

Moreover, since $r\ge 7$, there is at most one of $k_1,\cdots , k_s$, say $k_1$, that can be $\ge r-2$. If $k_1=r-2$, then

$$\begin{aligned} ||{\mathcal {A}}||_{L^2(\Omega ^{d_0})}&\le D_r ||{\mathfrak {D}}^{k_1}\partial \widehat{x}_{{}_{\!}I}||_{L^3(\Omega ^{d_0})}||\widehat{\partial }_{{}_{\!}I}{\mathfrak {D}}^{\ell } \widehat{\partial }_{{}_{\!}I}f_I-\widehat{\partial }_{{}_{\!}I{\!}I}{\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^6(\Omega ^{d_0})},\\ ||{\mathcal {B}}||_{L^2(\Omega ^{d_0})}&\le D_r ||{\mathfrak {D}}^{k_1}\partial \widehat{x}_{{}_{\!}I}-{\mathfrak {D}}^{k_1}\partial \widehat{x}_{{}_{\!}I{\!}I}||_{L^3(\Omega ^{d_0})}||\widehat{\partial }_{{}_{\!}I{\!}I}{\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^6(\Omega ^{d_0})}, \end{aligned}$$

and since $\ell \le 2$, we have

$$\begin{aligned}&||\widehat{\partial }_{{}_{\!}I}{\mathfrak {D}}^{\ell } \widehat{\partial }_{{}_{\!}I}\! f_I-\widehat{\partial }_{{}_{\!}I{\!}I}{\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}{}_{\!}||_{L^6(\Omega ^{d_0}\!)}\!\lesssim \! ||{{\,\mathrm{div}\,}}_{\!I}\! {\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I}\! f_{\!I}-{{\,\mathrm{div}\,}}_{{}_{\!}I{\!}I}\! {\mathfrak {D}}^{\ell } \widehat{\partial }_{{}_{\!}I{\!}I}{}_{\!} f_{{}_{\!}I{\!}I}{}_{\!}||_{L^6(\Omega ^{d_0}\!)}\\&\quad +||{{\,\mathrm{curl}\,}}_I \!{\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I}\!f_{\!I}-{{\,\mathrm{curl}\,}}_{{}_{\!}I{\!}I}\! {\mathfrak {D}}^{\ell } \widehat{\partial }_{{}_{\!}I{\!}I}{}_{\!} f_{{}_{\!}I{\!}I}{}_{\!}||_{L^6(\Omega ^{d_0}\!)}\,\,\,\\&\quad +||{\mathcal {T}^{}}{\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I}f_I-{\mathcal {T}^{}}{\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^6(\Omega ^{d_0})}+||\widehat{x}_{{}_{\!}I}\\&\quad -\widehat{x}_{{}_{\!}I{\!}I}||_{H^r(\Omega ^{d_0})}||\widehat{\partial }_{{}_{\!}I{\!}I}{\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^6(\Omega ^{d_0})} \lesssim ||{\mathfrak {D}}^\ell (g_I-g_{{}_{\!}I{\!}I})||_{L^6(\Omega ^{d_0})}\\&\quad +{\sum }_{\ell _1+\ell _2=\ell , \ell _1\ge 1}\Big (||({\mathfrak {D}}^{\ell _1}\widehat{A}_I)(\widehat{\partial }_{{}_{\!}I}{\mathfrak {D}}^{\ell _2}\widehat{\partial }_{{}_{\!}I}f_I-\widehat{\partial }_{{}_{\!}I{\!}I}{\mathfrak {D}}^{\ell _2}\widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I})||_{L^6(\Omega ^{d_0})}\\&\quad +||({\mathfrak {D}}^{\ell _1}[\widehat{A}_{{}_{\!}I}-\widehat{A}_{{}_{\!}I{\!}I}])(\widehat{\partial }_{{}_{\!}I{\!}I}{\mathfrak {D}}^{\ell _2}\widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I})||_{L^6(\Omega ^{d_0})}\Big )\\&\quad +||{\mathcal {T}^{}}{\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I}f_I-{\mathcal {T}^{}}{\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{H^1(\Omega ^{d_0})}+||\widehat{x}_{{}_{\!}I}-\widehat{x}_{{}_{\!}I{\!}I}||_{H^r(\Omega ^{d_0})}||\widehat{\partial }_{{}_{\!}I{\!}I}{\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^6(\Omega ^{d_0})}, \end{aligned}$$

where the sum is of lower order and

$$\begin{aligned}&||\partial _y({\mathcal {T}^{}}{\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I}f_I\!-{\mathcal {T}^{}}{\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I})||_{L^2(\Omega ^{d_0})}\lesssim ||\widehat{\partial }_{{}_{\!}I}{\mathcal {T}^{}}{\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I}{}_{\!}f_I\!\\&\quad -\widehat{\partial }_{{}_{\!}I{\!}I}{\mathcal {T}^{}}{\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})}+||(\widehat{\partial }_{{}_{\!}I{\!}I}\!-\!\widehat{\partial }_{{}_{\!}I}){\mathcal {T}^{}}{\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})}, \end{aligned}$$

which is of the form we control. Finally, if $k_1=r-1$, we need to control $\widehat{\partial }_{{}_{\!}I}^2 F-\widehat{\partial }_{{}_{\!}I{\!}I}^2 G$ in $L^\infty$, i.e.,

$$\begin{aligned}&||\widehat{\partial }_{{}_{\!}I}^2 f_I-\widehat{\partial }_{{}_{\!}I{\!}I}^2 f_{{}_{\!}I{\!}I}||_{L^\infty (\Omega ^{d_0})}\lesssim ||g_I-g_{{}_{\!}I{\!}I}||_{L^\infty (\Omega ^{d_0})}+||{\mathcal {T}^{}}(\widehat{\partial }_{{}_{\!}I}f_I-\widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I})||_{L^\infty (\Omega ^{d_0})}\\&\quad +||\widehat{x}_{{}_{\!}I}-\widehat{x}_{{}_{\!}I{\!}I}||_{H^r(\Omega ^{d_0})}||\widehat{\partial }_{{}_{\!}I{\!}I}^2 f_{{}_{\!}I{\!}I}||_{L^\infty (\Omega ^{d_0})}, \end{aligned}$$

where $||{\mathcal {T}^{}}(\widehat{\partial }_{{}_{\!}I}{}_{\!} f_I\!\!-\!\widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I})||_{L^{{}_{\!}\infty }(\Omega ^{d_0}\!)}\lesssim {\sum }_{\ell \le 1}||\partial _y^ \ell {\mathcal {T}^{}}(\widehat{\partial }_{{}_{\!}I}{}_{\!}f_I\!\!-\!\widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I})||_{L^{6_{\!}}(\Omega ^{d_0}\!)}$, and this can be controlled as above. $\quad \square$

Lemma C.8

Let $f_J=(g_J*\Phi )\circ \widehat{x}_J$ for $J=I,{}_{\!}I{\!}I$, where $g_J$ are smooth functions supported in $\Omega ^{d_0\!/2}$ satisfying ${\mathfrak {D}}g_J=0$ in $\Omega ^{d_0}\!\!\setminus \! \Omega$. Then

$$\begin{aligned} ||f_I-f_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})}\le D_r(||g_I-g_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})}+||\,\widetilde{\!x}_{{}_{\!}I}-\,\widetilde{\!x}_{{}_{\!}I\!I}||_{H^r(\Omega )}||g_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})}),\nonumber \\ \end{aligned}$$

(C.26)

and for $r\ge 7$, we have

$$\begin{aligned}&||D_t^{r-1} f_I-D_t^{r-1} f_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})}\le D_r\bigg ({\sum }_{k\le r-1}||{\mathfrak {D}}^k \widehat{\partial }_{{}_{\!}I}f_I-{\mathfrak {D}}^k\widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})}\nonumber \\&\quad +{\sum }_{k\le r-1}||{\mathfrak {D}}^k (g_I-g_{{}_{\!}I{\!}I})||_{L^2(\Omega ^{d_0})}\nonumber \\&\quad +{\sum }_{k\le 2}||{\mathfrak {D}}^k (g_I-g_{{}_{\!}I{\!}I})||_{L^6(\Omega ^{d_0})} +||g_I-g_{{}_{\!}I{\!}I}||_{L^\infty (\Omega ^{d_0})} +\big \{|| \,\widetilde{\!x}_{{}_{\!}I}-\,\widetilde{\!x}_{{}_{\!}I\!I}||_{H^r(\Omega )}\nonumber \\&\quad +{\sum }_{k\le r-2}||D_t^k (V_{{}_{\!}{}_{\!}I}-V_{{}_{\!}I{\!}I})||_{H^{r-k}(\Omega )}\big \}\nonumber \\&\quad \cdot \,\big ({\sum }_{k\le r\!-\!1}||{\mathfrak {D}}^k\widehat{\partial }_{{}_{\!}I{\!}I}\! f_{{}_{\!}I{\!}I\!}||_{L^2(\Omega ^{d_0}\!)}\nonumber \\&\quad +{\sum }_{k\le r\!-\!1}||{\mathfrak {D}}^k \! g_{{}_{\!}I{\!}I\!}||_{L^2(\Omega ^{d_0}\!)}\! \nonumber \\&\quad +{\sum }_{k\le 2}||{\mathfrak {D}}^k\! g_{{}_{\!}I{\!}I\!}||_{L^6(\Omega ^{d_0}\!)}\! +||g_{{}_{\!}I{\!}I\!}||_{L^\infty (\Omega ^{d_0}\!)}\big )\!\bigg ). \end{aligned}$$

(C.27)

Proof

We prove (C.26) first. Writing $f_J=\int _{\Omega ^{d_0}}g_J(t,y')\Phi (\widehat{x}_J(t,y)-\widehat{x}_J(t,y'))\widehat{\kappa }_J\,\mathrm{d}y$, we have

$$\begin{aligned} f_I-f_{{}_{\!}I{\!}I}= & {} \int _{\Omega ^{d_0}}\underbrace{g_{{}_{\!}I{\!}I}(t,y')\big (\Phi (\widehat{x}_{{}_{\!}I}(t,y)-\widehat{x}_{{}_{\!}I}(t,y')) -\Phi \big (\widehat{x}_{{}_{\!}I{\!}I}(t,y)-\widehat{x}_{{}_{\!}I{\!}I}(t,y')\big )\big )\widehat{\kappa }_{I}\,\mathrm{d}y'}_{I_1}\nonumber \\&+\!\!\int _{\Omega ^{d_0}}\underbrace{\!\!\!\!\!\!(g_I(t,y')\!-g_{{}_{\!}I{\!}I}(t,y')) \Phi \big (\widehat{x}_{{}_{\!}I}(t,y)\!-\widehat{x}_{{}_{\!}I}(t,y')\big )\widehat{\kappa }_{I}\mathrm{d}y'\!\!}_{I_2}\, \nonumber \\&\quad +\!\!\int _{\Omega ^{d_0}} \underbrace{\!\!\!\!\!g_{{}_{\!}I{\!}I}(t,y')\Phi \big (\widehat{x}_{{}_{\!}I{\!}I}(t,y)\!-\widehat{x}_{{}_{\!}I{\!}I}(t,y')\big ) (\widehat{\kappa }_{I}\!-\widehat{\kappa }_{{}_{\!}I{\!}I}) \mathrm{d}y'\!\!}_{I_3}. \end{aligned}$$

(C.28)

By Young’s inequality, we have

$$\begin{aligned} ||I_2||_{L^2(\Omega ^{d_0})}\le D_r||g_I-g_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})},\quad ||I_3||_{L^2(\Omega ^{d_0})}\le D_r ||\,\widetilde{\!x}_{{}_{\!}I}-\,\widetilde{\!x}_{{}_{\!}I\!I}||_{H^r(\Omega )}||g_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})}. \end{aligned}$$

To control $I_1$, we write

$$\begin{aligned}&\big |\Phi (\widehat{x}_{{}_{\!}I}(t,y)-\widehat{x}_{{}_{\!}I}(t,y'))-\Phi (\widehat{x}_{{}_{\!}I{\!}I}(t,y)-\widehat{x}_{{}_{\!}I{\!}I}(t,y'))\big |\\&\quad =\frac{1}{4\pi } \Big |\frac{1}{|\widehat{x}_I(t,y)-\widehat{x}_I(t,y')|}-\frac{1}{|\widehat{x}_{{}_{\!}I{\!}I}(t,y)-\widehat{x}_{{}_{\!}I{\!}I}(t,y')|} \Big | \\&\quad \le \frac{1}{4\pi }\frac{|\widehat{x}_{{}_{\!}I{\!}I}(t,y)-\widehat{x}_{{}_{\!}I}(t,y)|+|\widehat{x}_{{}_{\!}I{\!}I}(t,y')-\widehat{x}_{{}_{\!}I}(t,y')|}{|\widehat{x}_I(t,y)-\widehat{x}_I(t,y')||\widehat{x}_{{}_{\!}I{\!}I}(t,y)-\widehat{x}_{{}_{\!}I{\!}I}(t,y')|}. \end{aligned}$$

Since this is in $L^{\!1\!}(\Omega ^{d_0}\!)$, we have $||I_{\!1\!}||_{L^2(\Omega ^{d_0}\!)}\!\le \! D_r(||\,\widetilde{\!x}_{{}_{\!}I}\!||_{H^r(\Omega )})||\,\widetilde{\!x}_{{}_{\!}I}-\,\widetilde{\!x}_{{}_{\!}I\!I}||_{H^r(\Omega )}||g_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0}\!)}$ using Young’s inequality. Now, for (C.27), we write

$$\begin{aligned} D_t^{r-1}f_I&= (D_t^{r-1}g_I)*\Phi \circ \widehat{x}_{{}_{\!}I}+([\widehat{\Delta }_I, D_t^{r-1}]f_I)*\Phi \circ \widehat{x}_{{}_{\!}I},\\ D_t^{r-1}f_{{}_{\!}I{\!}I}&= (D_t^{r-1}g_{{}_{\!}I{\!}I})*\Phi \circ \widehat{x}_{{}_{\!}I{\!}I}+([\widehat{\Delta }_{{}_{\!}I{\!}I}, D_t^{r-1}]f_{{}_{\!}I{\!}I})*\Phi \circ \widehat{x}_{{}_{\!}I{\!}I}. \end{aligned}$$

To control $||D_t^{r-1} f_I\!-D_t^{r-1} f_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})}$, we need the bounds for $||(D_t^{r-1}\!g_I)*\Phi \circ \widehat{x}_{{}_{\!}I}-(D_t^{r-1}\!g_{{}_{\!}I{\!}I})*\Phi \circ \widehat{x}_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})}$ and $||([\widehat{\Delta }_I, D_t^{r-1}]f_I)*\Phi \circ \widehat{x}_{{}_{\!}I}-([\widehat{\Delta }_{{}_{\!}I{\!}I}, D_t^{r-1}]f_{{}_{\!}I{\!}I})*\Phi \circ \widehat{x}_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0}\!)}$. As above, they are controlled by

$$\begin{aligned}&D_r \big \{||D_t^{r-1}g_I-D_t^{r-1}g_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})} +||\,\widetilde{\!x}_{{}_{\!}I}-\,\widetilde{\!x}_{{}_{\!}I\!I}||_{H^r(\Omega )}||D_t^{r-1} g_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})}\big \},\\&D_r \big \{||[\widehat{\Delta }_I, D_t^{r-1}]f_I-[\widehat{\Delta }_{{}_{\!}I{\!}I}, D_t^{r-1}]f_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})}+||\,\widetilde{\!x}_{{}_{\!}I}-\,\widetilde{\!x}_{{}_{\!}I\!I}||_{H^r(\Omega )}||[\widehat{\Delta }_{{}_{\!}I{\!}I}, D_t^{r-1}]f_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})}\big \}, \end{aligned}$$

respectively, where $[\widehat{\Delta }_{{}_{\!}I{\!}I}, D_t^{r-1}]f_{{}_{\!}I{\!}I}$ can be treated by adapting the proof for Lemma C.4. Moreover, since for each $J=I, II$, $[\widehat{\Delta }_J, D_t^{r-1}]$ consists

$$\begin{aligned} (\widehat{\partial }_J^2 D_t^{\ell _1} \widehat{V}_J)\cdots (\widehat{\partial }_J D_t^{\ell _{n-1}} \widehat{V}_J)\cdot ( D_t^{\ell _n} \widehat{\partial }_J )\quad \text {and}\quad (\widehat{\partial }_J D_t^{\ell _1} \widehat{V}_J)\cdots (\widehat{\partial }_J D_t^{\ell _{n-1}} \widehat{V}_J)\cdot (\widehat{\partial }_J D_t^{\ell _n} \widehat{\partial }_J), \end{aligned}$$

where $\ell _1+\cdots +\ell _n=r-n$, the control of $||[\widehat{\Delta }_I, D_t^{r-1}]f_I-[\widehat{\Delta }_{{}_{\!}I{\!}I}, D_t^{r-1}]f_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})}$ requires that of

$$\begin{aligned}&K_1=||(\widehat{\partial }_{{}_{\!}I}^2 D_t^{\ell _1} \widehat{V}_{{}_{\!}I})\cdots (\widehat{\partial }_{{}_{\!}I}D_t^{\ell _{n-1}} \widehat{V}_{{}_{\!}I})\cdot ( D_t^{\ell _n} \widehat{\partial }_{{}_{\!}I}f_I )\\&\quad -(\widehat{\partial }_{{}_{\!}I{\!}I}^2 D_t^{\ell _1} \widehat{V}_{{}_{\!}I{\!}I})\cdots (\widehat{\partial }_{{}_{\!}I{\!}I}D_t^{\ell _{n-1}} \widehat{V}_{{}_{\!}I{\!}I})\cdot ( D_t^{\ell _n} \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I} )||_{L^2(\Omega ^{d_0})},\\&K_2=||(\widehat{\partial }_{{}_{\!}I}D_t^{\ell _1} \widehat{V}_{{}_{\!}I})\cdots (\widehat{\partial }_{{}_{\!}I}D_t^{\ell _{n-1}} \widehat{V}_{{}_{\!}I})\cdot ( \widehat{\partial }_{{}_{\!}I}D_t^{\ell _n} \widehat{\partial }_{{}_{\!}I}f_I )\\&\quad -(\widehat{\partial }_{{}_{\!}I{\!}I}D_t^{\ell _1} \widehat{V}_{{}_{\!}I{\!}I})\cdots (\widehat{\partial }_{{}_{\!}I{\!}I}D_t^{\ell _{n-1}} \widehat{V}_{{}_{\!}I{\!}I})\cdot ( \widehat{\partial }_{{}_{\!}I{\!}I}D_t^{\ell _n} \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I})||_{L^2(\Omega ^{d_0})}. \end{aligned}$$

It suffices to consider the case when $n=2$ only. To control $K_1$, we have, with $L^2=L^2(\Omega ^{d_0})$,

$$\begin{aligned}&||(\widehat{\partial }_{{}_{\!}I}^2 D_t^{\ell _1} \!\widehat{V}_{{}_{\!}I})D_t^{\ell _2} \widehat{\partial }_{{}_{\!}I}{}_{\!} {}_{\!} f_I\! -(\widehat{\partial }_{{}_{\!}I{\!}I}^2 D_t^{\ell _1}\! \widehat{V}_{{}_{\!}I{\!}I}) D_t^{\ell _2} \widehat{\partial }_{{}_{\!}I{\!}I}{}_{\!} f_{{}_{\!}I{\!}I}||_{L^2}\\&\quad \le ||(\widehat{\partial }_{{}_{\!}I}^2 D_t^{\ell _1}\! \widehat{V}_{{}_{\!}I}\!-\widehat{\partial }_{{}_{\!}I{\!}I}^2 D_t^{\ell _1}\! \widehat{V}_{{}_{\!}I{\!}I}) D_t^{\ell _2} \widehat{\partial }_{{}_{\!}I{\!}I}{}_{\!} f_{{}_{\!}I{\!}I} ||_{L^2}+||(\widehat{\partial }_{{}_{\!}I}^2 \! D_t^{\ell _1} \! \widehat{V}_{{}_{\!}I})(D_t^{\ell _2\!} \widehat{\partial }_{{}_{\!}I}{}_{\!}f_{\!I} \!-D_t^{\ell _2\!} \widehat{\partial }_{{}_{\!}I{\!}I}{}_{\!} f_{{}_{\!}I{\!}I})||_{L^2}\\&\quad \le \underbrace{||\big ((\widehat{\partial }_{{}_{\!}I}^2\!-\widehat{\partial }_{{}_{\!}I{\!}I}^2) D_t^{\ell _1} \!\widehat{V}_{{}_{\!}I}\!\big )D_t^{\ell _2} \widehat{\partial }_{{}_{\!}I{\!}I}{}_{\!}f_{{}_{\!}I{\!}I}||_{L^2}}_{K_{12}}+\underbrace{||\big (\widehat{\partial }_{{}_{\!}I{\!}I}^2 D_t^{\ell _1} \! (\widehat{V}_{{}_{\!}I}\!-\!\widehat{V}_{{}_{\!}I{\!}I})\big ) D_t^{\ell _2} \widehat{\partial }_{{}_{\!}I{\!}I}{}_{\!} f_{{}_{\!}I{\!}I} ||_{L^2}}_{K_{13}}\\&\qquad + \underbrace{||(\widehat{\partial }_{{}_{\!}I}^2\! D_t^{\ell _1} \! \widehat{V}_{{}_{\!}I})(D_t^{\ell _2} \widehat{\partial }_{{}_{\!}I}{}_{\!}f_{\!I} \!-\!D_t^{\ell _2\!} \widehat{\partial }_{{}_{\!}I{\!}I}{}_{\!} f_{{}_{\!}I{\!}I})||_{L^2}}_{K_{11}}\!. \end{aligned}$$

When $\ell _1\!\le \!r\!-\!4$, we bound $\widehat{V}\!$ factors in $L^{\!\infty \!}$ and use Sobolev’s lemma. Then $K_{11}\!\le \! D_r||D_t^{\ell _2} (\widehat{\partial }_{{}_{\!}I}{}_{\!}f_{\!I} \!- \widehat{\partial }_{{}_{\!}I{\!}I}{}_{\!} f_{{}_{\!}I{\!}I\!})||_{L^2}$, and $K_{12}\!\le \! D_r||\,\widetilde{\!x}_{{}_{\!}I}\!-\!\,\widetilde{\!x}_{{}_{\!}I\!I}||_{H^r(\Omega )}|| D_t^{\ell _2} \widehat{\partial }_{{}_{\!}I{\!}I}{}_{\!} f_{{}_{\!}I{\!}I}||_{L^2}$, and $K_{13}\!\le \! D_r ||D_t^{\ell _1}\!(V_{{}_{\!}I}\!-\!V_{{}_{\!}I{\!}I})||_{H^{r\!-\!\ell _1}} || D_t^{\ell _2} {}_{\!} \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^{\!2}}$. When $\ell _1\!=\!r\!-\!3$ (and $\ell _2\!=\!1$), we bound $\widehat{V}$ terms in $L^3(\Omega ^{d_0}\!)$ and use Sobolev’s lemma. In this case, $K_{11}\!\le \! D_r||D_t (\widehat{\partial }_{{}_{\!}I}{}_{\!} f_{\!I}\! - \widehat{\partial }_{{}_{\!}I{\!}I}{}_{\!} f_{{}_{\!}I{\!}I\!})||_{L^6(\Omega ^{d_0}\!)}$, and $K_{12}\!\le \! D_r||\,\widetilde{\!x}_{{}_{\!}I}\!-\!\,\widetilde{\!x}_{{}_{\!}I\!I}||_{H^r(\Omega )}|| D_t\widehat{\partial }_{{}_{\!}I{\!}I}{}_{\!} f_{{}_{\!}I{\!}I\!}||_{L^6(\Omega ^{d_0}\!)}$ and $K_{13}\le D_r ||D_t^{\ell _1}\!(V_{{}_{\!}I}\!-\!V_{{}_{\!}I{\!}I})||_{H^{r-3}(\Omega )} || D_t\widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^6(\Omega ^{d_0}\!)}$. By Sobolev’s lemma $||D_t (\widehat{\partial }_{{}_{\!}I}\! f_{\!I} -{}_{\!}\widehat{\partial }_{{}_{\!}I{\!}I}{}_{\!} f_{{}_{\!}I{\!}I})||_{L^6(\Omega ^{d_0}\!)}\!\lesssim \! ||D_t(\widehat{\partial }_{{}_{\!}I}f_{\!I \!}-\widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I})||_{H^{1\!}(\Omega ^{d_0}\!)},$ where we have $||\partial _y D_t (\widehat{\partial }_{{}_{\!}I}f_{\!I} - \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I})||_{L^2(\Omega ^{d_0})}\le D_r\big (||\widehat{\partial }_{{}_{\!}I}D_t \widehat{\partial }_{{}_{\!}I}{}_{\!}f_I\!-\widehat{\partial }_{{}_{\!}I{\!}I}D_t \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})}+||(\widehat{\partial }_{{}_{\!}I{\!}I}-\widehat{\partial }_{{}_{\!}I})D_t\widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})}\big ),$ which can be controlled by the right hand side of (C.27) using Lemma C.3, and $||D_t\widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^6(\Omega ^{d_0})}$ can be treated in a similar way. When $\ell _1\!=\!r\!-\!2$ (and $\ell _2\!=\!0$), we bound $\widehat{V}\!$ terms in $L^2(\Omega ^{d_0})$, so we need to control $||\widehat{\partial }_{{}_{\!}I}\! f_I\!-\widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^\infty (\Omega ^{d_0})}$ and $||\widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^\infty (\Omega ^{d_0})}$. By Sobolev’s lemma, $||\widehat{\partial }_{{}_{\!}I}f_I \!\!-\widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^\infty (\Omega ^{d_0})} \lesssim {\sum }_{\ell \le 1}||\partial _y^\ell (\widehat{\partial }_{{}_{\!}I}f_I\! -\widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I})||_{L^6(\Omega ^{d_0})},$ where $||\partial _y(\widehat{\partial }_{{}_{\!}I}f_I\! -\widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I})||_{L^6(\Omega ^{d_0})}\le ||\widehat{\partial }_{{}_{\!}I}^2 f_I\!-\widehat{\partial }_{{}_{\!}I{\!}I}^2 f_{{}_{\!}I{\!}I}||_{L^6(\Omega ^{d_0})}+||(\widehat{A}_{{}_{\!}I{\!}I}\!-\widehat{A}_{I})\partial _y \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^6(\Omega ^{d_0})},$ which is of the form that we control thanks to Lemma C.3, and $||\widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^\infty (\Omega ^{d_0})}$ can be treated in a similar fashion. Finally, we control $K_2$ by adapting a similar argument as above. $\quad \square$

Lemma C.9

Let $g_J$, $J=I,{}_{\!}I{\!}I$ be smooth functions supported in $\Omega ^{d_0\!/2}$ and $y\in \partial \Omega ^{d_0}$. Then

$$\begin{aligned}&|\partial _y {\mathfrak {D}}^k(f_I(t,y)-f_{{}_{\!}I{\!}I}(t,y)) | \lesssim ||{\mathfrak {D}}^k(g_I-g_{{}_{\!}I{\!}I})||_{L^2(\Omega ^{d_0})}\\&\quad +||\,\widetilde{\!x}_{{}_{\!}I}-\,\widetilde{\!x}_{{}_{\!}I\!I}||_{H^r(\Omega )}||{\mathfrak {D}}^k g_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})} ,\quad k\ge 0. \end{aligned}$$

Proof

Since $\widehat{x}(t,y)=x_0(y)$ and $\widehat{V}(t,y)=0$ when $y\in \partial \Omega ^{d_0}$, we have that $[\widetilde{\Delta }_{{}_{\!}I}, {\mathfrak {D}}^k]f_I(y)=[\widetilde{\Delta }_{{}_{\!}I{\!}I}, {\mathfrak {D}}^k]f_{{}_{\!}I{\!}I}(y)=0$. Therefore, $\partial _y{\mathfrak {D}}^k f_I(t,y)=({\mathfrak {D}}^k g_I)*(\partial _y \Phi )(t,y)$ and $\partial _y{\mathfrak {D}}^k f_{{}_{\!}I{\!}I}(t,y)=({\mathfrak {D}}^k g_{{}_{\!}I{\!}I})*(\partial _y \Phi )(t,y)$, and the control of $|\partial _y {\mathfrak {D}}^k(f_I(t,y)-f_{{}_{\!}I{\!}I}(t,y))|$ follows from a similar argument that is used to control (C.28) since $\partial _y\Phi (\widehat{x}(t,y)-\widehat{x}(t,y'))$ is away from its singularity when $y'\in \Omega ^{d_0/2}$ and $y\in \partial \Omega ^{d_0}$. $\quad \square$

Theorem C.10

With the same assumptions as in Lemma C.8, if $r\ge 7$, we have, with $L^p=L^p(\Omega ^{d_0})$,

$$\begin{aligned}&{\sum }_{k\le r-1}||{\mathfrak {D}}^k \widehat{\partial }_{{}_{\!}I}f_I-{\mathfrak {D}}^k\widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^2} \le D_r\bigg ({\sum }_{k\le r-1}||{\mathfrak {D}}^k (g_I-g_{{}_{\!}I{\!}I})||_{L^2} \nonumber \\&\quad +{\sum }_{k\le 2}||{\mathfrak {D}}^k (g_I-g_{{}_{\!}I{\!}I})||_{L^6} +||g_I-g_{{}_{\!}I{\!}I}||_{L^\infty }\,\,\nonumber \\&\quad +\!\big (||\,\widetilde{\!x}_{{}_{\!}I}\!{}_{\!}-\!\,\widetilde{\!x}_{{}_{\!}I\!I}\!||_{H^{r\!}(\Omega )}\!+\!\!{\sum }_{k\le r-2}||D_t^k (V_{{}_{\!}{}_{\!}I}\!-\!V_{{}_{\!}I{\!}I}\!)||_{H^{r\!-\!k}(\Omega )}\big ) \big ({\sum }_{k\le r\!-\!1}||{\mathfrak {D}}^k\! g_{{}_{\!}I{\!}I}\!||_{L^2}\nonumber \\&\quad \!+\!\! {\sum }_{k\le 2}||{\mathfrak {D}}^k\! g_{{}_{\!}I{\!}I}\!||_{L^6}\!+\!||g_{{}_{\!}I{\!}I}\!||_{L^\infty }\!\big )\!\bigg ). \end{aligned}$$

(C.29)

Proof

When $k=0$, this is done as in the proof of Lemma B.13. However, one needs to estimate $||f_I-f_{{}_{\!}I{\!}I}||_{L^2}$ directly without using Poincaré’s inequality, which has been done in Lemma C.8. Next, suppose that (C.29) is known for $k=0,\cdots ,r-2$. When $k=r-1$, we have

$$\begin{aligned}&||{\mathfrak {D}}^{r-1}\widehat{\partial }_{{}_{\!}I}f_I-{\mathfrak {D}}^{r-1}\widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})}^2 \nonumber \\&\quad =\int _{\Omega ^{d_0}} \!\! \delta ^{ij}\!\underbrace{\big ({\mathfrak {D}}^{r-1} \widehat{\partial }_{{}_{\!}I}{}_{\!i} f_I-{\mathfrak {D}}^{r-1}\widehat{\partial }_{{}_{\!}I{\!}I}{}_{\!i} f_{{}_{\!}I{\!}I}\big )\big (\widehat{\partial }_{{}_{\!}I}{}_{\!j} {\mathfrak {D}}^{r-1} f_I-\widehat{\partial }_{{}_{\!}I{\!}I}{}_{\!j} {\mathfrak {D}}^{r-1} f_{{}_{\!}I{\!}I}\big )\,\mathrm{d}y}_{I}\nonumber \\&\quad +\int _{\Omega ^{d_0}} \!\!\delta ^{ij}\big ({\mathfrak {D}}^{r-1} \widehat{\partial }_{{}_{\!}I}{}_{\!i} f_I-{\mathfrak {D}}^{r-1}\widehat{\partial }_{{}_{\!}I{\!}I}{}_{\!i} f_{{}_{\!}I{\!}I}\big )\big ([{\mathfrak {D}}^{r-1}, \widehat{\partial }_{{}_{\!}I}{}_{\!j}] f_I-[{\mathfrak {D}}^{r-1},\widehat{\partial }_{{}_{\!}I{\!}I}{}_{\!j}] f_{{}_{\!}I{\!}I}]\big )\,\mathrm{d}y. \nonumber \\ \end{aligned}$$

(C.30)

The second term can be bounded using Lemma D.3 together with the bounds for $||\widehat{\partial }_{{}_{\!}I}f_{\!I}\! -\!\widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^\infty (\Omega ^{d_0})}$ and ${\sum }_{\ell \le 2}||{\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I}{}_{\!} f_{\!I}\!-\!{\mathfrak {D}}^\ell {}_{\!}\widehat{\partial }_{{}_{\!}I{\!}I}{}_{\!}f_{{}_{\!}I{\!}I}||_{L^6(\Omega ^{d_0})}$. Here, $||\widehat{\partial }_{{}_{\!}I}{}_{\!}f_I \!-\widehat{\partial }_{{}_{\!}I{\!}I}{}_{\!}f_{{}_{\!}I{\!}I}||_{L^\infty (\Omega ^{d_0})} \lesssim {\sum }_{\ell \le 1}||\partial _y^\ell (\widehat{\partial }_{{}_{\!}I}{}_{\!}f_{\!I}\! -{}_{\!}\widehat{\partial }_{{}_{\!}I{\!}I}{}_{\!} f_{{}_{\!}I{\!}I})||_{L^6(\Omega ^{d_0})},$ where

$$\begin{aligned}&||\partial _y(\widehat{\partial }_{{}_{\!}I}f_I -\widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I})||_{L^6(\Omega ^{d_0})}\le ||\widehat{\partial }_{{}_{\!}I}^2 f_I-\widehat{\partial }_{{}_{\!}I{\!}I}^2 f_{{}_{\!}I{\!}I}||_{L^6(\Omega ^{d_0})}\\&\quad +||(\widehat{A}_{{}_{\!}I{\!}I\, i}^{\,\,\,a}-\widehat{A}_{{}_{\!}I\, i}^{\,\,a})\partial _{y^a} \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^6(\Omega ^{d_0})}, \end{aligned}$$

which is of the form that we control by Lemma C.7. In addition, for each $\ell \le 2$, we have with $L^p=L^p(\Omega ^{d_0})$:

$$\begin{aligned}&||{\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I}{}_{\!} f_{\!I\!}-{\mathfrak {D}}^\ell {}_{\!} \widehat{\partial }_{{}_{\!}I{\!}I}{}_{\!} f_{{}_{\!}I{\!}I}\!||_{L^6}\lesssim ||\partial _y({\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I}\! f_{\!I}\!-{\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I{\!}I}{}_{\!} f_{{}_{\!}I{\!}I})||_{L^2} \\&\quad \le ||\widehat{\partial }_{{}_{\!}I}{\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I}f_I\!-\widehat{\partial }_{{}_{\!}I{\!}I}{\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}\!||_{L^2}+||(\widehat{A}_{{}_{\!}I{\!}I\, i}^{\,\,\,a}\!-\widehat{A}_{{}_{\!}I\, i}^{\,\,a})\partial _{a} {\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I{\!}I}f_{{}_{\!}I{\!}I}||_{L^2}, \end{aligned}$$

which is again of the form that we control by Lemma C.7. To deal with the first term in (C.30), one writes ${\mathfrak {D}}^{r-1}\widehat{\partial }_{{}_{\!}I{\!}I}{}_{\!i} f_{{}_{\!}I{\!}I}\!=\!{\mathfrak {D}}^{r-1}\widehat{\partial }_{{}_{\!}I}{}_{\!i} f_{{}_{\!}I{\!}I}+{\mathfrak {D}}^{r-1}[(\widehat{A}_{{}_{\!}I{\!}I\, i}^{\,\,\,a}\!-\widehat{A}_{{}_{\!}I\, i}^{\,\,a})\partial _{a}f_{{}_{\!}I{\!}I}]$ and $\widehat{\partial }_{{}_{\!}I{\!}I}{}_{\!i}{\mathfrak {D}}^{r-1} f_{{}_{\!}I{\!}I} \!=\!\widehat{\partial }_{{}_{\!}I}{}_{\!i}{\mathfrak {D}}^{r-1} f_{{}_{\!}I{\!}I}+(\widehat{A}_{{}_{\!}I{\!}I\, i}^{\,\,\,a}\!-\widehat{A}_{{}_{\!}I\, i}^{\,\,a})\partial _{a}{\mathfrak {D}}^{r-1} f_{{}_{\!}I{\!}I}$, and

$$\begin{aligned}&I\!=\!\int _{\Omega ^{d_0}}\!\!\!\! \delta ^{ij}\big (\widehat{\partial }_{{}_{\!}I}{}_{\!i} {\mathfrak {D}}^{r-1} (f_I\!-f_{{}_{\!}I{\!}I})\big )\big ({\mathfrak {D}}^{r-1} \widehat{\partial }_{{}_{\!}I}{}_{\!j} (f_I\!- f_{{}_{\!}I{\!}I})\big )\mathrm{d}y \\&\quad + \int _{\Omega ^{d_0}}\!\!\!\! \delta ^{ij}\big (\widehat{\partial }_{{}_{\!}I}{}_{\!i} {\mathfrak {D}}^{r-1} (f_I\!-f_{{}_{\!}I{\!}I})\big )\big ({\mathfrak {D}}^{r-1} \big [(\widehat{A}_{{}_{\!}I{\!}I\, j}^{\,\,\,a}-\widehat{A}_{{}_{\!}I\, j}^{\,\,a})\partial _{a} f_{{}_{\!}I{\!}I}\big ]\big )\mathrm{d}y\\&\quad +\!\int _{\Omega ^{d_0}}\!\!\!\!\!\!\delta ^{ij}\big ((\widehat{A}_{{}_{\!}I{\!}I\, i}^{\,\,\,a}\!-\widehat{A}_{{}_{\!}I\, i}^{\,\,a})\partial _{a} {\mathfrak {D}}^{r-1\!} f_{{}_{\!}I{\!}I}\!\big )\big ({\mathfrak {D}}^{r\!-\!1\!} \widehat{\partial }_{{}_{\!}I}{}_{\!j} (f_{\!I\!}- f_{{}_{\!}I{\!}I}\!)\big )\mathrm{d}y \\&\quad +\!\int _{\Omega ^{d_0}}\!\!\!\!\!\!\delta ^{ij}\big ((\widehat{A}_{{}_{\!}I{\!}I\, i}^{\,\,\,a}\!-\widehat{A}_{{}_{\!}I\, i}^{\,\,a})\partial _{a} {\mathfrak {D}}^{r\!-\!1\!} f_{{}_{\!}I{\!}I}\!\big )\big ({\mathfrak {D}}^{r\!-\!1\!} \big [(\widehat{A}_{{}_{\!}I{\!}I\, j}^{\,\,\,a}-\widehat{A}_{{}_{\!}I\, j}^{\,\,a})\partial _{a} f_{{}_{\!}I{\!}I}\!\big ]\big )\mathrm{d}y. \end{aligned}$$

It is straightforward to control the last three terms, and the first term is equal to

$$\begin{aligned}&\int _{\Omega ^{d_0}}\!\!\!\! \delta ^{ij}\partial _{a}\big ((\widehat{A}_{{}_{\!}I\, i}^{\,\,a} {\mathfrak {D}}^{r\!-\!1} \! (f_{\!I}\!- \! f_{{}_{\!}I{\!}I})\big ){\mathfrak {D}}^{r\!-\!1} \widehat{\partial }_{{}_{\!}I}{}_{\!j} (f_{\!I}\!-\!f_{{}_{\!}I{\!}I})\big ) \mathrm{d}y\nonumber \\&\quad -\!\int _{\Omega ^{d_0}}\!\!\!\!\delta ^{ij} (\partial _{a}\widehat{A}_{{}_{\!}I\, i}^{\,\,a})\big ( {\mathfrak {D}}^{r\!-\!1 \!} (f_{\!I}\!- \! f_{{}_{\!}I{\!}I})\big )\big ({\mathfrak {D}}^{r-1\!} \widehat{\partial }_{{}_{\!}I}{}_{\!j} (f_{\!I}\!-\! f_{{}_{\!}I{\!}I})\big )\,\mathrm{d}y.\,\,\, \end{aligned}$$

(C.31)

Integrating the first term by parts gives:

$$\begin{aligned}&\int _{\Omega ^{d_0}} \delta ^{ij}\underbrace{\big ( {\mathfrak {D}}^{r-1} \!( f_I\!- f_{{}_{\!}I{\!}I})\big )\widehat{\partial }_{{}_{\!}I}{}_{\!i}\big ({\mathfrak {D}}^{r-1\!} \widehat{\partial }_{{}_{\!}I}{}_{\!j} (f_I\!- f_{{}_{\!}I{\!}I})\big )\, \mathrm{d}y}_{II}\\&\quad +\!\!\int _{\partial \Omega ^{d_0}}\delta ^{ij}\underbrace{N_a \widehat{A}_{{}_{\!}I{}_{\,} i}^{\,\,a} \big ( {\mathfrak {D}}^{r-1\!} (f_I\!- f_{{}_{\!}I{\!}I})\big )\big ({\mathfrak {D}}^{r-1} \widehat{\partial }_{{}_{\!}I}{}_{\!j} (f_I\!- f_{{}_{\!}I{\!}I})\big )\, \mathrm{d}y}_{{\mathcal {B}}}. \end{aligned}$$

Here, modulo controllable error terms, II is equal to $\int _{\Omega ^{d_0}} ({\mathfrak {D}}^r f_I-{\mathfrak {D}}^r f_{{}_{\!}I{\!}I})({\mathfrak {D}}^r g_I-{\mathfrak {D}}^r g_{{}_{\!}I{\!}I})\,\mathrm{d}y.$ When ${\mathfrak {D}}^{r-1}$ contains at least one ${\mathcal {T}^{}}$, one can integrate this ${\mathcal {T}^{}}$ by parts and control the resulting integral as what is done to the control of (C.14) in the proof of Theorem 7.2. When ${\mathfrak {D}}^{r-1}=D_t^{r-1}$, this is bounded by $||D_t^{r-1} f_I-D_t^{r-1} f_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})}||D_t^{r-1} g_I-D_t^{r-1} g_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})}$, where $||D_t^{r-1} f_I-D_t^{r-1} f_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})}$ can be controlled by Lemma C.8. The second term in (C.31) can be controlled in a similar way. On the other hand, since $\widehat{A}_{{}_{\!}I{}_{\,} i}^{\,\,a}=\delta ^{a}_i$ on $\partial \Omega ^{d_0}$, ${\mathcal {B}}$ can be controlled appropriately using the Lemma C.9. $\quad \square$

Appendix D: Estimates for Commutators and $F$

In this section, we fix a vector field $V = V(t,y)$ on $\Omega$. We let x(t, y) denote the flow of V(t, y), i.e. $D_t x=V$, $x|_{t=0}=x_0$, and let $\widetilde{x}(t,y)$ denote the tangentially smoothed flow, as in (4.1). We suppose that the mapping $y \mapsto \widetilde{x}(t,y)$ is invertible for each t, and we let $A^i_{\,\,a}$ and $A^a_{\,\,i}$ be the Jacobian matrix of $\widetilde{x}$ and its inverse, respectively, see (4.2). We will assume that $\widetilde{x}$ and V satisfy the bounds (5.1).

If $M^i_{\,\,a}$ is an invertible matrix with inverse $N_{\,\,i}^a$, we recall the formula for the derivatives of $N_{\,\,i}^a$:

$$\begin{aligned} D N_{\,\,i}^a = - N_{\,\,i}^b N_{\,\,j}^a \big (D M^j_{\,\,b}\big ), \end{aligned}$$

where here $D = D_t$ or $D = \partial _c$. When $M^i_{\,\,a} = A^i_{\,\,a}$, then this gives

$$\begin{aligned} D_t A^a_{\,\,i} = - A_{\,\,j}^a \big (\widetilde{\partial }_i S_\varepsilon V^j\big ),&\partial _c A^a_{\,\,i} = -A_{\,\,j}^a \big (\widetilde{\partial }_i u^j_c\big ). \end{aligned}$$

(D.1)

Using these formulas it is straightforward to calculate the following commutators:

$$\begin{aligned}{}[D_t, \widetilde{\partial }_i]&= -A_{\,\,i}^b A_{\,\,j}^a \big (\partial _b S_\varepsilon V^j\big ) \partial _a = -\big (\widetilde{\partial }_i S_\varepsilon V^j\big ) \widetilde{\partial }_j, \nonumber \\ [\partial _c, \widetilde{\partial }_i]&= -A_{\,\,i}^b A_{\,\,j}^a \big (\partial _c A^j_{\,\,b}\big ) \partial _a =- \big (\widetilde{\partial }_i A_{\,\,c}^j\big ) \widetilde{\partial }_j. \end{aligned}$$

(D.2)

We will need estimates for higher order derivatives of $A_{\,\,i}^a$. As in Section 3.3, given a set $\mathcal {V}= \{T_1,..., T_N\}$ of vector fields, we write $\mathcal {V}^r = \mathcal {V}\times \cdots \times \mathcal {V}$ (r times) as well as $\mathcal {V}^r V:\Omega \rightarrow \mathbb {R}^{3N+3}$. The families of vector fields we will consider are $\mathcal {V}= {\mathcal {T}^{}}$ (tangential derivatives, $\mathcal {V}= {\mathfrak {D}}$ (mixed tangential and time derivatives), $\mathcal {V}= \mathcal {D}$ (mixed full space and time derivatives), and $\mathcal {V}= \{\partial _y\}$. The point of the below estimate is just that derivatives of A behave like derivatives of $\partial _y \widetilde{x}$. This lemma is in fact essentially the same as Lemma D.5 but it is convenient to note this estimate separately.

Lemma D.1

With notation as in Section 3.3, if $T^I \in \mathcal {V}^s$ where $\mathcal {V}= {\mathcal {T}^{}}$,${\mathfrak {D}}$,$\mathcal {D}$, or $\{\partial _y\}$, then

$$\begin{aligned} ||T^I\! A_{\,\,i}^a||_{L^2} + ||T^I\!\! g^{ab}||_{L^2} \le C(M)\big (||T^I \widetilde{x}||_{H^{1}} + P(||\mathcal {V}^{s-2} \widetilde{x}||_{H^2})\big ) \end{aligned}$$

(D.3)

We note that taking $\mathcal {V}= \{\partial _y\}$ and summing over all $T^I \in \mathcal {V}^s$ gives

$$\begin{aligned} ||A_{\,\,i}^a||_{H^s} + ||g^{ab}||_{H^s(\Omega )} \le C(M)\big ( ||\widetilde{x}||_{H^{s+1}} + P( ||\widetilde{x}||_{H^{s}})\big ). \end{aligned}$$

(D.4)

Proof

The estimates for g follow from the estimates for A and the definition $g^{ab} = \delta ^{ij} A_{\,\,i}^a A_{\,\,j}^b$ so we just prove the estimates for A. For the sake of simplicity we will assume that all $T \in \mathcal {V}$ commutes with $\partial _y$; this is only not the case if $\mathcal {V}= {\mathcal {T}^{}}$ and in that case the commutator is lower order and can be handled using similar arguments to the below. For $T^I \in \mathcal {V}^s$, repeatedly applying (D.1), we have

$$\begin{aligned} T^{I} A_{\,\,i}^a = -A_{\,\,i}^b A_{\,\,j}^a T^I \partial _b \widetilde{x}^j - {\sum } \big (\widetilde{\partial }T^{I_1} \widetilde{x}\big ) \cdots \big (\widetilde{\partial }T^{I_k} \widetilde{x}\big ) \end{aligned}$$

(D.5)

where the sum is taken over a collection of multi-indices $I_1,..., I_k$ with $|I_1| + \cdots + |I_k| = s$ with $|I_j| \le s-1$ for $j = 1,..., k$. The first term is bounded by the first term on the right-hand side of (D.4). When $s \le 3$, we bound the first $k-1$ factors in each summand in $L^\infty$ by C(M) and the remaining factor in $L^2$ by $||\mathcal {V}^{s-1} \widetilde{x}||_{H^1}$ and this is bounded by the right-hand side of (D.4) for all the values of $\mathcal {V}$ we are considering. We now assume that $s \ge 4$. If any index $|I_j| \le \min (3, s-3)$, we use the Sobolev estimate (A.23) to bound $||\widetilde{\partial }T^{I_{\!j\!}} x||_{L^{\!\infty }}\! \le C(_{{}_{\!}}M_{{}_{\!}}) ||\partial _y T^{I_{\!j\!}} x||_{L^{\!\infty }}\! \le C(_{{}_{\!}}M_{{}_{\!}}) ||\mathcal {V}^{s-\!2} x||_{H^2}$. Therefore it suffices to deal with the case when at least one index $|I_j| \ge \max (4, s-4)$. There can be at most one such index because if there are $\ell \ge 2$ such terms then $4\ell \le s$ so that $s \ge 8$ and that $\ell (s-4) \le s$ so that $s \le 4$. Since there is one such index and $|I_j| \le s-1$ we bound the corresponding term in $L^2$ by $||\mathcal {V}^{s-1} \widetilde{x}||_{H^1}$ which completes the proof. $\quad \square$

Similarly, we have

Lemma D.2

Define $\,\widetilde{\!x}_{{}_{\!}I}, \,\widetilde{\!x}_{{}_{\!}I\!I}, A_I, A_{{}_{\!}I{\!}I}, \widetilde{g}_{{}_{\!}I}, \widetilde{g}_{{}_{\!}I{\!}I}$ as in Appendix B. With notation as in Lemma D.1, if $T^I \in \mathcal {V}^s$:

$$\begin{aligned}&||T^I\!(_{\!}A_{{}_{\!}I\, i}^{\,\,a}\! -\! A_{{}_{\!}I{\!}I\, i}^{\,\,\,a})||_{L^2}\! + \! ||T^I\!(\widetilde{g}_{{}_{\!}I}^{ab}\!\!- \!\widetilde{g}_{{}_{\!}I{\!}I}^{ab})||_{L^2} \! \le \mathcal {D}_{\!s} ||\,\widetilde{\!x}_{{}_{\!}I}\! - \,\widetilde{\!x}_{{}_{\!}I\!I}\!||_{H^{\ell +1}},\nonumber \\&\qquad \mathcal {D}_{\!s} \!=\! \mathcal {D}_{\!s}(M{}_{\!},_{\!} ||\mathcal {V}^{s-\!1\!}\,\widetilde{\!x}_{{}_{\!}I}||_{H^{2}}, ||\mathcal {V}^{s-\!1\!}\,\widetilde{\!x}_{{}_{\!}I\!I}||_{H^{2}}). \end{aligned}$$

(D.6)

Proof

Applying (D.1) to $A_{{}_{\!}I\, a}^{\,\,i}$ and $A_{{}_{\!}I{\!}I\, a}^{\,\,\,i}$ generates two sums of the form (D.5). Subtracting these two sums and arguing as in the proof of the previous lemma gives (D.6). $\quad \square$

The next lemma will be used at several places. Recall the definitions of $\Omega ^{d_0}$, $\widehat{\partial }_{{}_{\!}I}, \widehat{\partial }_{{}_{\!}I{\!}I}$ from Section 7.1.

Lemma D.3

Let with $r\ge 5$. Then there is a continuous function

$$\begin{aligned} C_r = C_r\big (M', ||\,\widetilde{\!x}_{{}_{\!}I}||_{H^r(\Omega )}, ||\,\widetilde{\!x}_{{}_{\!}I\!I}||_{H^r(\Omega )}, {{{\,\mathrm{{\textstyle {\sum }}}\,}}}_{\ell \le r-1}||D_t^\ell V_{{}_{\!}{}_{\!}I}||_{H^{r-\ell }(\Omega )}, {{{\,\mathrm{{\textstyle {\sum }}}\,}}}_{\ell \le r-1}||D_t^\ell V_{{}_{\!}I{\!}I}||_{H^{r-\ell }(\Omega )}\big ), \end{aligned}$$

such that with ${\mathfrak {D}}^r$ the mixed space-time tangential derivatives defined in Section 3.3,

$$\begin{aligned}&||[{\mathfrak {D}}^r,\widehat{\partial }_{{}_{\!}I}] f - [{\mathfrak {D}}^r, \widehat{\partial }_{{}_{\!}I{\!}I}] g||_{L^2(\Omega ^{d_0}\!)}\\&\quad \le C_r \bigg ( {\sum }_{\ell \le r-1} ||{\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I}f - {\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I{\!}I}g||_{L^2(\Omega ^{d_0}\!)} + ||{\mathcal {T}^{}}\,\widetilde{\!x}_{{}_{\!}I}||_{H^r(\Omega )}\big \{||\widehat{\partial }_{{}_{\!}I}f - \widehat{\partial }_{{}_{\!}I{\!}I}g||_{L^\infty (\Omega ^{d_0}\!)} \\&\quad + {\sum }_{\ell \le 2} ||{\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I}f - {\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I{\!}I}g||_{L^6(\Omega ^{d_0}\!)}\big \}\\&\quad + \big \{||{\mathcal {T}^{}}\,\widetilde{\!x}_{{}_{\!}I}- {\mathcal {T}^{}}\,\widetilde{\!x}_{{}_{\!}I\!I}||_{H^r(\Omega )} +{\sum }_{\ell \le r-1}||D_t^{\ell }\big ((S_\varepsilon V_{{}_{\!}I} -S_\varepsilon V_{{}_{\!}I{\!}I}\big )||_{H^{r-\ell }(\Omega )}\big \} (||\widehat{\partial }_{{}_{\!}I{\!}I}g||_{L^\infty (\Omega ^{d_0}\!)}\\&\quad +{\sum }_{\ell \le 2}||{\mathfrak {D}}^{\ell }\widehat{\partial }_{{}_{\!}I{\!}I}g||_{L^6(\Omega ^{d_0}\!)})\\&\quad +\big \{||\,\widetilde{\!x}_{{}_{\!}I}-\,\widetilde{\!x}_{{}_{\!}I\!I}||_{H^r(\Omega )} \\&\quad +{\sum }_{\ell \le r-1}||D_t^{\ell }\big (S_\varepsilon V_{{}_{\!}I} - S_\varepsilon V_{{}_{\!}I{\!}I}\big )||_{H^{r-\ell }(\Omega )}\big \} {\sum }_{\ell \le r-1}||{\mathfrak {D}}^\ell \widehat{\partial }_{{}_{\!}I{\!}I}g||_{L^2(\Omega ^{d_0}\!)}\bigg ). \end{aligned}$$

Proof

We start by writing

$$\begin{aligned}&{[}{\mathfrak {D}}^r,\widehat{\partial }_{{}_{\!}I}] f - [{\mathfrak {D}}^r, \widehat{\partial }_{{}_{\!}I{\!}I}] g = \underbrace{-\big ((\widehat{\partial }_{{}_{\!}I}{\mathfrak {D}}^r\widehat{x}_{{}_{\!}I})\widehat{\partial }_{{}_{\!}I}f-(\widehat{\partial }_{{}_{\!}I{\!}I}{\mathfrak {D}}^r\widehat{x}_{{}_{\!}I{\!}I})\widehat{\partial }_{{}_{\!}I{\!}I}g\big )}_{I}\\&\quad +{\sum }_{\ell _1+\cdots +\ell _s=r,\ell _i\le r-1}\underbrace{ (\partial {\mathfrak {D}}^{\ell _1}\widehat{x}_{{}_{\!}I})\cdots (\partial {\mathfrak {D}}^{\ell _{s-1}}\widehat{x}_{{}_{\!}I})({\mathfrak {D}}^{\ell _s}\widehat{\partial }_{{}_{\!}I}f) -(\partial {\mathfrak {D}}^{\ell _1}\widehat{x}_{{}_{\!}I{\!}I})\cdots (\partial {\mathfrak {D}}^{\ell _{s-1}}\widehat{x}_{{}_{\!}I{\!}I})({\mathfrak {D}}^{\ell _s}\widehat{\partial }_{{}_{\!}I{\!}I}g)}_{II}. \end{aligned}$$

We have

$$\begin{aligned}&||I||_{L^2(\Omega ^{d_0})}\le ||(\widehat{\partial }_{{}_{\!}I}{\mathfrak {D}}^r\widehat{x}_{{}_{\!}I}-\widehat{\partial }_{{}_{\!}I{\!}I}{\mathfrak {D}}^r\widehat{x}_{{}_{\!}I{\!}I})\widehat{\partial }_{{}_{\!}I{\!}I}g||_{L^2(\Omega ^{d_0})}+||(\widehat{\partial }_{{}_{\!}I}{\mathfrak {D}}^r\widehat{x}_{{}_{\!}I})(\widehat{\partial }_{{}_{\!}I}f-\widehat{\partial }_{{}_{\!}I{\!}I}g)||_{L^2(\Omega ^{d_0})}\\&\quad \le ||\widehat{\partial }_{{}_{\!}I}{\mathfrak {D}}^r\widehat{x}_{{}_{\!}I}-\widehat{\partial }_{{}_{\!}I{\!}I}{\mathfrak {D}}^r\widehat{x}_{{}_{\!}I{\!}I}||_{L^2(\Omega ^{d_0})}||\widehat{\partial }_{{}_{\!}I{\!}I}g||_{L^\infty (\Omega ^{d_0})}+ ||\widehat{\partial }_{{}_{\!}I}{\mathfrak {D}}^r\widehat{x}_{{}_{\!}I}||_{L^2(\Omega ^{d_0})}||\widehat{\partial }_{{}_{\!}I}f-\widehat{\partial }_{{}_{\!}I{\!}I}g||_{L^\infty (\Omega ^{d_0})}\\&\quad \le \bigg (||{\mathcal {T}^{}}(\widehat{x}_{{}_{\!}I}-\widehat{x}_{{}_{\!}I{\!}I})||_{H^r(\Omega ^{d_0})}+{\sum }_{\ell \le r-1}||D_t^{\ell }(\widehat{V}_{{}_{\!}I}-\widehat{V}_{{}_{\!}I{\!}I})||_{H^{r-\ell }(\Omega ^{d_0})}\bigg )||\widehat{\partial }_{{}_{\!}I{\!}I}g||_{L^\infty (\Omega ^{d_0})}\\&\quad +\bigg (||{\mathcal {T}^{}}\widehat{x}_{{}_{\!}I}||_{H^r(\Omega ^{d_0})}+{\sum }_{\ell \le r-1}||D_t^{\ell }\widehat{V}_{{}_{\!}I}||_{H^{r-\ell }(\Omega ^{d_0})}\bigg )||\widehat{\partial }_{{}_{\!}I}f-\widehat{\partial }_{{}_{\!}I{\!}I}g||_{L^\infty (\Omega ^{d_0})}. \end{aligned}$$

In addition, to control II in $L^2$ one only needs to consider

$$\begin{aligned} II_1= & {} (\partial {\mathfrak {D}}^{\ell _1}\widehat{x}_{{}_{\!}I})\cdots (\partial {\mathfrak {D}}^{\ell _{s-1}}\widehat{x}_{{}_{\!}I})({\mathfrak {D}}^{\ell _s}\widehat{\partial }_{{}_{\!}I}f-{\mathfrak {D}}^{\ell _s}\widehat{\partial }_{{}_{\!}I{\!}I}g), \\ II_2= & {} (\partial {\mathfrak {D}}^{\ell _1}\widehat{x}_{{}_{\!}I}-\partial {\mathfrak {D}}^{\ell _1}\widehat{x}_{{}_{\!}I{\!}I})(\partial {\mathfrak {D}}^{\ell _2}\widehat{x}_{{}_{\!}I{\!}I})\cdots (\partial {\mathfrak {D}}^{\ell _{s-1}}\widehat{x}_{{}_{\!}I{\!}I}){\mathfrak {D}}^{\ell _s}\widehat{\partial }_{{}_{\!}I{\!}I}g. \end{aligned}$$

When $r\!-\!1\!\ge \! \ell _s\!\ge \! 3$ then $\ell _j\!\le \! r\!-\!3$ for $j\!\le \!s\!-\!1$ and we control the terms involving $\widehat{x}$ in $L^\infty$. Hence

$$\begin{aligned} ||II_1||_{L^2(\Omega ^{d_0})}\le C_r'||{\mathfrak {D}}^{\ell _s}\widehat{\partial }_{{}_{\!}I}f- {\mathfrak {D}}^{\ell _s}\widehat{\partial }_{{}_{\!}I{\!}I}g||_{L^2(\Omega ^{d_0})}, \end{aligned}$$

where

$$\begin{aligned}&C_r' = C_r'\big (M', ||\widehat{x}_{{}_{\!}I}||_{H^r(\Omega ^{d_0})}, ||\widehat{x}_{{}_{\!}I{\!}I}||_{H^r(\Omega ^{d_0})}, {{{\,\mathrm{{\textstyle {\sum }}}\,}}}_{\ell \le r-1}||D_t^\ell \widehat{V}_{{}_{\!}I}||_{H^{r-\ell }(\Omega ^{d_0})},\\&\quad {{{\,\mathrm{{\textstyle {\sum }}}\,}}}_{\ell \le r-1}||D_t^\ell \widehat{V}_{{}_{\!}I{\!}I}||_{H^{r-\ell }(\Omega ^{d_0})}\big ), \end{aligned}$$

and

$$\begin{aligned} ||II_2||_{L^2(\Omega ^{d_0})}\le & {} C_r' ||\partial {\mathfrak {D}}^{\ell _1}\widehat{x}_{{}_{\!}I}-\partial {\mathfrak {D}}^{\ell _1}\widehat{x}_{{}_{\!}I{\!}I}||_{L^\infty (\Omega ^{d_0})}||{\mathfrak {D}}^{\ell _s}\widehat{\partial }_{{}_{\!}I{\!}I}g||_{L^2(\Omega ^{d_0})}\\\le & {} C_r'\bigg ( ||\widehat{x}_{{}_{\!}I}- \widehat{x}_{{}_{\!}I{\!}I}||_{H^r(\Omega ^{d_0})}\\&+{\sum }_{\ell \le r-1}||D_t^{\ell }(\widehat{V}_{{}_{\!}I}- \widehat{V}_{{}_{\!}I{\!}I})||_{H^{r-\ell }(\Omega ^{d_0})}\bigg )||{\mathfrak {D}}^{\ell _s}\widehat{\partial }_{{}_{\!}I{\!}I}g||_{L^2(\Omega ^{d_0})}. \end{aligned}$$

Second, when $\ell _j\ge r-2$ for $j=1,\cdots , s-1$, since $r\ge 5$, there is at most one $\ell _j$, say $\ell _1$, can be greater than or equal to $r-2$. In this case, we have

$$\begin{aligned}&||II_1||_{L^2(\Omega ^{d_0})}\le C_r'||\partial {\mathfrak {D}}^{\ell _1}\widehat{x}_{{}_{\!}I}||_{L^3(\Omega ^{d_0})}||{\mathfrak {D}}^{\ell _s}\widehat{\partial }_{{}_{\!}I}f-{\mathfrak {D}}^{\ell _s}\widehat{\partial }_{{}_{\!}I{\!}I}g||_{L^6(\Omega ^{d_0})} \\&\quad \le C_r'\bigg (||{\mathcal {T}^{}}\widehat{x}_I||_{L^2(\Omega ^{d_0})}+{\sum }_{\ell \le r-1}||D_t^{\ell }\widehat{V}_{{}_{\!}I}||_{H^{r-\ell }(\Omega ^{d_0})}\bigg )||{\mathfrak {D}}^{\ell _s}\widehat{\partial }_{{}_{\!}I}f-{\mathfrak {D}}^{\ell _s}\widehat{\partial }_{{}_{\!}I{\!}I}g||_{L^6(\Omega ^{d_0})}, \end{aligned}$$

where $\ell _s\le 2$, and

$$\begin{aligned}&||II_2||_{L^2(\Omega ^{d_0})}\le C_r'||\partial {\mathfrak {D}}^{\ell _1}\widehat{x}_{{}_{\!}I}-\partial {\mathfrak {D}}^{\ell _1}\widehat{x}_{{}_{\!}I{\!}I}||_{L^3(\Omega ^{d_0})}||{\mathfrak {D}}^{\ell _s}\widehat{\partial }_{II} g||_{L^6(\Omega ^{d_0})}\\&\quad \le C_r'\bigg ( ||{\mathcal {T}^{}}(\widehat{x}_{{}_{\!}I}- \widehat{x}_{{}_{\!}I{\!}I})||_{H^r(\Omega ^{d_0})}\\&\quad +{\sum }_{\ell \le r-1}||D_t^{\ell }(\widehat{V}_{{}_{\!}I}- \widehat{V}_{{}_{\!}I{\!}I})||_{H^{r-\ell }(\Omega ^{d_0})}\bigg )||{\mathfrak {D}}^{\ell _s}\widehat{\partial }_{II} g||_{L^6(\Omega ^{d_0})}. \end{aligned}$$

This concludes the proof after adapting the Sobolev extension theorem. $\quad \square$

As a consequence, if we take $g = 0$, $\,\widetilde{\!x}_{{}_{\!}I}= \,\widetilde{\!x}_{{}_{\!}I\!I}\equiv \widetilde{x}$, we have

Corollary D.4

If $r\!\ge \! 5$ then there is a constant $C_r\! =\! C_r(M_0, ||\widetilde{x}||_{H^r(\Omega )}, {\sum }_{\ell \le r-1}||D_t^\ell V||_{H^{r-\ell }(\Omega )})$ such that

$$\begin{aligned}&||[{\mathfrak {D}}^r,\widehat{\partial }] f||_{L^2(\Omega ^{d_0})}\le C_r\bigg (||{\mathcal {T}^{}}\widetilde{x}||_{H^r(\Omega )}\big (||\widehat{\partial }f||_{L^\infty (\Omega ^{d_0})} +{\sum }_{\ell \le 2}||{\mathfrak {D}}^\ell \widehat{\partial }f||_{L^6(\Omega ^{d_0})}\big ) \nonumber \\&\quad +{\sum }_{\ell \le r-1}||{\mathfrak {D}}^\ell \widehat{\partial }f||_{L^2(\Omega ^{d_0})}\bigg ). \end{aligned}$$

Here ${\mathfrak {D}}^r$ be the mixed space-time tangential derivatives defined in Section 3.3. In particular, one has

$$\begin{aligned}&||[{\mathfrak {D}}^{r-1}D_t,\widehat{\partial }] f||_{L^2(\widehat{\mathcal {D}}_t)}\le C_r\bigg (||\widehat{\partial }f||_{L^\infty (\widehat{\mathcal {D}}_t)}+{\sum }_{\ell \le 2}||{\mathfrak {D}}^\ell \widehat{\partial }f||_{L^6(\widehat{\mathcal {D}}_t)} \\&\quad +{\sum }_{\ell \le r-1}||{\mathfrak {D}}^\ell \widehat{\partial }f||_{L^2(\widehat{\mathcal {D}}_t)}\bigg ). \end{aligned}$$

The following lemma is similar to the previous one but is better adapted to proving estimates for the wave equation. As in the previous lemma, the point is that the commutator between r derivatives and $\widetilde{\partial }$ is a differential operator of order r with coefficients depending on $r+1$ derivatives of $\widetilde{x}$.

In the next lemma, we will assume that we have the following a priori bound for $\,\widetilde{\!x}_{{}_{\!}I}, \,\widetilde{\!x}_{{}_{\!}I\!I}$:

$$\begin{aligned} {{{\,\mathrm{{\textstyle {\sum }}}\,}}}_{|I| + k \le 3} |D_t^k \partial _y^I \,\widetilde{\!x}_{{}_{\!}I}| + |D_t^k \partial _y^I \,\widetilde{\!x}_{{}_{\!}I\!I}| \le M. \end{aligned}$$

(D.7)

If we are considering vector fields which do not involve time derivatives, we can instead assume that only

$$\begin{aligned} {{{\,\mathrm{{\textstyle {\sum }}}\,}}}_{|I| \le 3} |\partial _y^I \,\widetilde{\!x}_{{}_{\!}I}| + |D_t^k \partial _y^I \,\widetilde{\!x}_{{}_{\!}I\!I}| \le M_0. \end{aligned}$$

(D.8)

Lemma D.5

Fix $s \ge 0$ and suppose that (D.7) holds. If $\mathcal {V}= \mathcal {D}, {\mathfrak {D}}$ or $\mathcal {V}= {\mathcal {T}^{}}$, there is a constant $C_s = C_s(M, ||\mathcal {V}^{s-2} \,\widetilde{\!x}_{{}_{\!}I}||_{H^2(\Omega )}, ||\mathcal {V}^{s-2} \,\widetilde{\!x}_{{}_{\!}I\!I}||_{H^2(\Omega )})$ so that if $T^J \in \mathcal {V}^s$, with notation as in Section 3.3, then

$$\begin{aligned}&||[T^J, \widetilde{\partial }_{{}_{\!}I}] f - [T^J, \widetilde{\partial }_{{}_{\!}I{\!}I}] g||_{L^2} \le C_s\big ( ||T^J \,\widetilde{\!x}_{{}_{\!}I\!I}||_{H^1} + 1\big ) {\sum }_{j \le s} ||\mathcal {V}^{j-2} (\widetilde{\partial }_{{}_{\!}I}f - \widetilde{\partial }_{{}_{\!}I{\!}I}g)||_{H^1} \nonumber \\&\quad + C_s\big ( |||T^J(\,\widetilde{\!x}_{{}_{\!}I}- \,\widetilde{\!x}_{{}_{\!}I\!I})||_{H^1} + ||\mathcal {V}^{s-1} (\,\widetilde{\!x}_{{}_{\!}I}- \,\widetilde{\!x}_{{}_{\!}I\!I})||_{H^1}\nonumber \\&\quad + ||\mathcal {V}^{2} (\,\widetilde{\!x}_{{}_{\!}I}- \,\widetilde{\!x}_{{}_{\!}I\!I})||_{C^1_{y,t}}\big ) {\sum }_{j \le s} ||\mathcal {V}^{j-2} \widetilde{\partial }_{{}_{\!}I{\!}I}g||_{H^1}, \end{aligned}$$

(D.9)

with $H^s = H^s(\Omega )$ and $||\alpha ||_{C^1_{y,t}} = \sum _{k + |J| \le 1} ||\partial _y^J D_t^k \alpha ||_{L^\infty ([0,T] \times \Omega )}$. If $\mathcal {V}= \{\partial _{y^1}, \partial _{y^2}, \partial _{y^3}\}$, the above estimate holds assuming (D.8) holds with M replaced by $M_0$.

Before proving this lemma we record a few useful instances of it which will be used at several points. Taking $g= 0$ and writing $\widetilde{x}=\,\widetilde{\!x}_{{}_{\!}I}$, with $C_s' =C_s'(M, ||{\mathcal {T}^{}}^{s-2} \widetilde{x}||_{H^2(\Omega )})$, we have

$$\begin{aligned} ||[\partial _y^I, \widetilde{\partial }] f||_{L^2(\Omega )}\le & {} C_s(M_0, ||\widetilde{x}||_{H^{s+1}}) ||\widetilde{\partial }f||_{H^{s-1}(\Omega )}, \quad |I| = s,\\ ||[T^J, \widetilde{\partial }]f||_{L^2(\Omega )}\le & {} C_s' (||T^J \widetilde{x}||_{H^1(\Omega )} + 1){\sum }_{j \le s-1}||{\mathcal {T}^{}}^{j}\widetilde{\partial }f||_{H^1(\Omega )}, \quad T^J \!\!\in {\mathcal {T}^{}}^s\! . \end{aligned}$$

Proof

Using (D.1), we have

$$\begin{aligned}&{[}T^J,\widetilde{\partial }_{{}_{\!}I}] f - [T^J, \widetilde{\partial }_{{}_{\!}I{\!}I}] g = \underbrace{-\big ((\widetilde{\partial }_{{}_{\!}I}T^J\,\widetilde{\!x}_{{}_{\!}I})\widetilde{\partial }_{{}_{\!}I}f-( T^J \,\widetilde{\!x}_{{}_{\!}I\!I})\widetilde{\partial }_{{}_{\!}I{\!}I}g\big )}_{I} \nonumber \\&\quad +{\sum }_{J_1+\cdots +J_m = J, |J_i| \le s-1}\underbrace{ (\partial T^{J_1}\,\widetilde{\!x}_{{}_{\!}I})\cdots (\partial T^{J_{m-1}}\,\widetilde{\!x}_{{}_{\!}I})T^{J_m}\widetilde{\partial }_{{}_{\!}I}f -(\partial T^{J_1}\,\widetilde{\!x}_{{}_{\!}I\!I})\cdots (\partial T^{J_{m-1}}\,\widetilde{\!x}_{{}_{\!}I\!I})T^{J_m}\widetilde{\partial }_{{}_{\!}I{\!}I}g}_{II}, \nonumber \\ \end{aligned}$$

(D.10)

We control

$$\begin{aligned}&||(\widetilde{\partial }_{{}_{\!}I}{}_{\!} T^J \!\,\widetilde{\!x}_{{}_{\!}I})\widetilde{\partial }_{{}_{\!}I}\! f{}_{\!} - (\widetilde{\partial }_{{}_{\!}I{\!}I}{}_{\!} T^J\! \,\widetilde{\!x}_{{}_{\!}I\!I})\widetilde{\partial }_{{}_{\!}I{\!}I}g||_{L^2(\Omega )} \le || (\widetilde{\partial }_{{}_{\!}I}{}_{\!} T^J\! \,\widetilde{\!x}_{{}_{\!}I}- \widetilde{\partial }_{{}_{\!}I{\!}I}{}_{\!} T^J \!\,\widetilde{\!x}_{{}_{\!}I\!I})\widetilde{\partial }_{{}_{\!}I{\!}I}g||_{L^2(\Omega )} \nonumber \\&\quad + ||(\widetilde{\partial }_{{}_{\!}I{\!}I}{}_{\!} T^J \!\,\widetilde{\!x}_{{}_{\!}I\!I}) (\widetilde{\partial }_{{}_{\!}I}\! f \!- \widetilde{\partial }_{{}_{\!}I{\!}I}g)||_{L^2(\Omega )}. \end{aligned}$$

(D.11)

If $|J| \le 2$, then we control the factors involving $\,\widetilde{\!x}_{{}_{\!}I}, \,\widetilde{\!x}_{{}_{\!}I\!I}$ in $L^\infty$ and the result is bounded by the right-hand side of (D.9). If instead $|J| \ge 3$, we control the factors involving f, g in $L^\infty$ and note that since we must have $s \ge 3$, by the Sobolev estimate (A.23), we have

$$\begin{aligned} ||\widetilde{\partial }_{{}_{\!}I{\!}I}g||_{L^\infty (\Omega )} \le C{\sum }_{j \le 2} ||\mathcal {V}^j \widetilde{\partial }_{{}_{\!}I{\!}I}g||_{H^1(\Omega )} \le C{\sum }_{j \le s} ||\mathcal {V}^j \widetilde{\partial }_{{}_{\!}I{\!}I}g||_{H^1(\Omega )}, \end{aligned}$$

which is bounded by the right-hand side of (D.9). Bounding $||\widetilde{\partial }_{{}_{\!}I}f - \widetilde{\partial }_{{}_{\!}I{\!}I}g||_{L^\infty (\Omega )}$ in the same way shows that the left-hand side of (D.11) is controlled by the right-hand side of (D.9).

To control II, it suffices to consider

$$\begin{aligned}&II_1\!=\!(\partial T^{J_1\!}\,\widetilde{\!x}_{{}_{\!}I}-\partial T^{J_1\!}\,\widetilde{\!x}_{{}_{\!}I\!I}\!)(\partial T^{J_2\!}\,\widetilde{\!x}_{{}_{\!}I\!I}\!){}_{\!}\cdots {}_{\!}(\partial T^{J_{m\!-\!1}\!}\,\widetilde{\!x}_{{}_{\!}I\!I}\!) T^{J_m}\widetilde{\partial }_{{}_{\!}I{\!}I}g,\\&\qquad II_2\!=\!(\partial T^{J_1\!}\,\widetilde{\!x}_{{}_{\!}I}\!){}_{\!}\cdots {}_{\!}(\partial T^{J_{m\!-\!1}\!}\,\widetilde{\!x}_{{}_{\!}I}\!)T^{J_m\!}(\widetilde{\partial }_{{}_{\!}I}\! f\!- \widetilde{\partial }_{{}_{\!}I{\!}I}g), \end{aligned}$$

where $J_1\! +\! ... + \!J_m \!=\! J$ and $|J_{\!1\!}|,..., |J_{m\!}| \le \!s\!-\!1$. We will just bound $||II_2||_{L^2(\Omega )}$, since the estimate for $||II_{1}||_{L^2(\Omega )}$ is similar. We start by noting that for each $J_i$ with $|J_i| \le 2, i \le m-1$, we control the corresponding factors of $\,\widetilde{\!x}_{{}_{\!}I}$ in $L^\infty$ by the right-hand side of (D.9). Rearranging indices, it therefore suffices to control

$$\begin{aligned}&||(\widetilde{\partial }_{{}_{\!}I}T^{J_1} \,\widetilde{\!x}_{{}_{\!}I})\cdots (\widetilde{\partial }_{{}_{\!}I}T^{J_\ell } \,\widetilde{\!x}_{{}_{\!}I}) (T^{J_m} \widetilde{\partial }_{{}_{\!}I}(f-g))||_{L^2(\Omega )},\\&\quad |J_1|, ..., |J_\ell | \ge 3, |J_1| + \cdots + |J_\ell | + |J_m| \le s-1. \end{aligned}$$

If there are no factors of $\,\widetilde{\!x}_{{}_{\!}I}$ present then the result is bounded by $||\widetilde{\partial }T^{J_m} (f-g)||_{L^2(\Omega )}$ and since $J_m \le s-1$ we control this by the right-hand side of (D.9). If there is at least one factor of $\,\widetilde{\!x}_{{}_{\!}I}$ present, Note that the conditions on the $|J_k|$ force $|J^m| \le s-4$ and so we control the last factor in $L^\infty$ by $\sum _{j \le s-2} ||\mathcal {V}^j \widetilde{\partial }(f-g)||_{H^1(\Omega )}$ by the Sobolev estimate (A.23). We now use Holder’s inequality and the Sobolev embedding (A.19) to control

$$\begin{aligned}&||\widetilde{\partial }_{{}_{\!}I}T^{J_{{}_{\!}1}}\,\widetilde{\!x}_{{}_{\!}I}\!\cdots \widetilde{\partial }_{{}_{\!}I}T^{J_\ell } \,\widetilde{\!x}_{{}_{\!}I}||_{L^{{}_{\!}2}(\Omega )} \!\le C ||\widetilde{\partial }_{{}_{\!}I}T^{J_1}\,\widetilde{\!x}_{{}_{\!}I}||_{L^{{}_{\!}2\ell }(\Omega )} \!\cdots ||\widetilde{\partial }_{{}_{\!}I}T^{J_\ell }\,\widetilde{\!x}_{{}_{\!}I}||_{L^{{}_{\!}2\ell }(\Omega )} \\&\quad \!\le C ||\widetilde{\partial }_{{}_{\!}I}T^{J_1}\,\widetilde{\!x}_{{}_{\!}I}||_{H^{1{}_{\!}}(\Omega )} \!\cdots ||\widetilde{\partial }_{{}_{\!}I}T^{J_\ell } \,\widetilde{\!x}_{{}_{\!}I}||_{H^{1{}_{\!}}(\Omega )}. \end{aligned}$$

We now note that since $|J_1| + ... + |J_\ell | \le s-1$ and $|J_k| \ge 3$ for each $k = 1,..., \ell$, we in fact have $|J_k| \le s-2$ for each k, and so each of these factors is controlled by the right-hand side of (D.9). $\quad \square$

We also need a version with pure time derivatives in the proof of the estimates for the wave equation.

Lemma D.6

Fix $s \ge 0$. If (5.2) holds, there is a constant $C_s = C_s(M, ||\widetilde{x}||_s, ||V||_{\mathcal {X}^s})$ so that

$$\begin{aligned} || [D_t^{s+1}, \widetilde{\partial }] f||_{L^2(\Omega )} \le C_s (||\widetilde{\partial }f||_{s,0} + (||V||_{\mathcal {X}^{s+1}} + 1)||\widetilde{\partial }f||_{s-1}). \end{aligned}$$

(D.12)

Proof

By (D.10) with $g = 0$ and $T^J = D_t^{s+1}$, we have

$$\begin{aligned}&[D_t^{s+1}, \widetilde{\partial }_i] f = -(\widetilde{\partial }D_t^{s+1} \widetilde{x})(\widetilde{\partial }f) +{\sum }_{s_1 + ... + s_m = s+1,\,\, s_m\ge 1}\\&\quad (\partial D_t^{s_1} \widetilde{x}) \cdots (\partial D_t^{s_{m-1}}\widetilde{x}) (D_t^{s_m} \widetilde{\partial }f). \end{aligned}$$

We now argue as in the previous lemma, but we want to point out explicitly how the norms of V arise. We write the first term as $-(\widetilde{\partial }D_t^s S_\varepsilon V)(\widetilde{\partial }f)$. If $s \le 2$ then we control the first factor in $L^\infty$ since $S_\varepsilon : L^\infty \rightarrow L^\infty$. If instead $s \ge 3$, we control the second factor using Sobolev embedding, $||\widetilde{\partial }f||_{L^\infty (\Omega )} \le C||\widetilde{\partial }f||_{H^2(\Omega )} \le C||\widetilde{\partial }f||_{s-1}$, and now we note that $||\widetilde{\partial }D_t^s S_\varepsilon V||_{L^2(\Omega )} \le C(M) ||V||_{\mathcal {X}^{s+1}}$, using that $S_\varepsilon :L^2\rightarrow L^2$.

The terms in the sum can be controlled using essentially the same argument as in the previous lemma. Rearranging indices it suffices to control

$$\begin{aligned}&||(\widetilde{\partial }D_t^{s_1}\widetilde{x})\cdots (\widetilde{\partial }D_t^{s_{j}} \widetilde{x}) (D_t^{s_m}\widetilde{\partial }f)||_{L^2(\Omega )},\\&\quad s_\ell \ge 3, \ell = 1,..., j, s_1 + \cdots s_j + s_m \le s, s_m \ge 1. \end{aligned}$$

If there are no factors of $\widetilde{x}$ present then we control this by $||D_t^{s_m}\widetilde{\partial }f||_{L^2(\Omega )} \le ||\widetilde{\partial }f||_{s,0}$. and if there is at least one factor of $\widetilde{x}$ present then we must have $s_m \le s-3$ and so we can control $||D_t^{s_m}\widetilde{\partial }f||_{L^\infty (\Omega )} \le C||D_t^{s_m}\widetilde{\partial }f||_{H^2(\Omega )} \le C||\widetilde{\partial }f||_{s-1}$. When $j = 1$ the result is obvious since $||\widetilde{\partial }D_t^{s_1} \widetilde{x}||_{L^2(\Omega )} \le C||\widetilde{x}||_{s_1 + 1}$ and $s_1 \le s-1$. When $j \ge 2$ we have, by Sobolev embedding (A.19),

$$\begin{aligned}&||(\widetilde{\partial }D_t^{s_1}\widetilde{x})\cdots (\widetilde{\partial }D_t^{s_j} \widetilde{x})||_{L^2(\Omega )} \le C ||\widetilde{\partial }D_t^{s_1}\widetilde{x}||_{L^{2j}(\Omega )} \cdots ||\widetilde{\partial }D_t^{s_j}\widetilde{x}||_{L^{2j}(\Omega )} \\&\quad \le C ||\widetilde{\partial }D_t^{s_1}\widetilde{x}||_{H^1(\Omega )} \cdots ||\widetilde{\partial }D_t^{s_j} \widetilde{x}||_{H^1(\Omega )}. \end{aligned}$$

Since each $s_\ell$ must satisfy $s_\ell \le s-3$, each of these factors is bounded by $C(M)||\widetilde{x}||_{s-1}$, as required. $\quad \square$

We also need to use the following commutator estimates in $\widetilde{\mathcal {D}}_t$.

Lemma D.7

Let $r \ge 7$ and $k+\ell \le r+1$ with $k\ge 2$, we have

$$\begin{aligned} ||[D_t^{k-1}, \widetilde{\partial }]\varphi ||_{H^\ell (\mathcal {D}_t)} \le P\big ({{{\,\mathrm{{\textstyle {\sum }}}\,}}}_{s\le k-2}||D_t^sS_\varepsilon V||_{H^{r-s}(\widetilde{\mathcal {D}}_t)}\big ){\sum }_{s\le k-2}||D_t^s\varphi ||_{H^{r-s}(\widetilde{\mathcal {D}}_t)}, \end{aligned}$$

(D.13)

and for $k+\ell =r$, we have

$$\begin{aligned} ||[D_t^{k-1}, \widetilde{\Delta }]\varphi ||_{H^\ell (\mathcal {D}_t)} \le P\big ({{{\,\mathrm{{\textstyle {\sum }}}\,}}}_{s\le k-2}||D_t^sS_\varepsilon V||_{H^{r-s}(\widetilde{\mathcal {D}}_t)}\big ){\sum }_{s\le k-2}||D_t^s\varphi ||_{H^{r-s}(\widetilde{\mathcal {D}}_t)}. \end{aligned}$$

(D.14)

Proof

It is not hard to compute that $[D_t^{k-1}, \widetilde{\partial }]$ consists of terms of the following forms:

$$\begin{aligned} (\widetilde{\partial }D_t^{s_1} S_\varepsilon V)\cdots (\widetilde{\partial }D_t^{s_{n-1}} S_\varepsilon V)(\widetilde{\partial }D_t^{s_n} \varphi ),\quad \text {with}\quad s_1\!+\cdots +s_n=k-n,\,\,\,n \ge 2, \end{aligned}$$

so (D.13) follows. On the other hand, (D.14) follows that $[D_t^{k-1}\!\!, \widetilde{\Delta }]$ consists of terms of the following form:

$$\begin{aligned}&(\widetilde{\partial }D_t^{s_3} S_\varepsilon V)\cdots (\widetilde{\partial }D_t^{s_n} S_\varepsilon V)(\widetilde{\partial }^2 D_t^{s_1} S_\varepsilon V)(\widetilde{\partial }D_t^{s_2} \varphi ), \quad \text {with}\quad {s_1+\cdots +s_n=k-n,\,\,\,n\ge 2},\\&(\widetilde{\partial }D_t^{s_3} S_\varepsilon V)\cdots (\widetilde{\partial }D_t^{s_n} S_\varepsilon V)(\widetilde{\partial }D_t^{s_1} S_\varepsilon V)(\widetilde{\partial }^2 D_t^{s_2} \varphi ), \quad \text {with}\quad {s_1+\cdots +s_n=k-n,\,\,\,n\ge 2}.\qquad \end{aligned}$$

$\square$

We now prove some estimates which are used in Sects. 6 and 8 to control the terms on the right-hand side of the various wave equations. For these estimates we will assume the following bound for $\varphi$:

$$\begin{aligned} {{{\,\mathrm{{\textstyle {\sum }}}\,}}}_{k + |J| \le 3} |D_t^k \partial _y^J \widetilde{\partial }\varphi | + |D_t^k \varphi | \le L. \end{aligned}$$

(D.15)

Lemma D.8

If the equation of state satisfies (1.9) for all $j \ge k + \ell \equiv s$, then there is a constant C depending only on $c_1, c_2$ and a polynomial P so that

$$\begin{aligned} || D_t^{k} (e'(\varphi )D_t^2\varphi ) - e'(\varphi ) (D_t^{k+2} \varphi )||_{H^\ell } \le C L ||D_t^{k+1} \varphi ||_{H^{\ell }} + P(L, ||\varphi ||_{s}). \end{aligned}$$

(D.16)

Proof

We just prove the $\ell \!=\! 0$ case since $\ell \!\ge \!1$ is similar. The main term in $D_t^{k} (e^{\prime \!}(\varphi )D_t^2\varphi ) - e^{\prime \!}(\varphi ) (D_t^{k+2} \varphi )$ is

$$\begin{aligned} e''(\varphi ) (D_t \varphi )D_t^{k+1} \varphi , \end{aligned}$$

(D.17)

and the remaining terms are of the form

$$\begin{aligned}&e^{(m)}(\varphi )D_t^{k_1}\varphi \cdots D_t^{k_m}\varphi ,\quad 2\le m\le k, \quad k_1+\cdots +k_m = k+1,\nonumber \\&\quad \text { where } k_j \le k, 1 \le j \le m. \end{aligned}$$

(D.18)

The term (D.17) is bounded by the right-hand side of (D.16). To bound (D.18), we note that if $k_j \le 3$ for $1 \le j \le m$ all of the terms are bounded by L. If there are any terms with $k_j \le k-2$ then by Sobolev embedding we have $||D_t^{k_j} \varphi ||_{L^\infty } \le C ||\varphi ||_k$. Therefore it just remains to consider the case that there is at least one j with $k_j \ge \max (4, k-1)$ and in fact there can be at most one such term since we also have $k_j \le k$ for each j. In this case we put the corresponding factor in $L^2$ and this proves (D.16). $\quad \square$

The following estimate is nearly the same as (D.16) but will be used in Section F.2 to bound quantities of the form $e'(f) g$ when we know that f is smoother than g.

Lemma D.9

Under the hypotheses of Lemma D.8, if $k + \ell = s$ then there is a constant C depending only on $c_1, c_2$ on is a polynomial P so that if $\varphi$ satisfies (D.15),

$$\begin{aligned} ||D_t^k e'(\varphi )||_{H^\ell } \le C \big ( ||D_t^k \varphi ||_{H^\ell } + P(L, ||\varphi ||_{s-1})). \end{aligned}$$

Proof

The proof is similar to the proof of Lemma D.8. If $|I|\! =\! \ell$ then $\partial _y^I D_t^k e'(\varphi )$ is a sum of terms

$$\begin{aligned} e^{m}(\varphi ) (\partial _y^{I_1}D_t^{k_1}\varphi )\cdots (\partial _y^{I_m} D_t^{k_m}\varphi ) \varphi ),\qquad |I_1| + ... + |I_m| =|I|,\quad k_1 + ... + k_m=k. \end{aligned}$$

Using Sobolev embedding as in the proof of Lemma D.8 gives the result. $\quad \square$

We will also need estimates for the derivatives of $F= F^1 + F^2$, where

$$\begin{aligned} F^1 = -(\widetilde{\partial }_i S_\varepsilon V^j)(\widetilde{\partial }_j V^i), \qquad F^2 = - e''(h) (D_t h)^2-\rho (h). \end{aligned}$$

Lemma D.10

If (5.2) holds and h satisfies (D.15), then for $k \ge 1$,

$$\begin{aligned} ||D_t^k F_1||_{L^2}&\le C(M, ||\partial V||_{L^\infty })\big (||D_t^k V||_{H^1} + P(||V||_{\mathcal {X}^k})\big ), \end{aligned}$$

(D.19)

$$\begin{aligned} ||D_t^k F_2||_{L^2}&\le C L ||D_t^{k+1} h||_{L^2} + P(L, ||h||_{k,0}). \end{aligned}$$

(D.20)

For $k \ge 0$, writing $k + \ell = s$, we also have

$$\begin{aligned} ||D_t^k F_1||_{H^\ell }&\le C(M,||\partial V||_{L^\infty })\big ( ||D_t^k V||_{H^{\ell +1}} + ||\widetilde{x}||_{H^{\ell +1}} + P(||V||_{s}, ||\widetilde{x}||_{H^{\ell }})\big ), \end{aligned}$$

(D.21)

$$\begin{aligned} ||D_t^k F_2||_{H^\ell }&\le CL ||D_t^{k+1} h||_{H^{\ell }} + P(L, ||h||_{s}). \end{aligned}$$

(D.22)

Proof

We we just prove the estimate (D.21). The estimate (D.19) follows in a similar manner and the estimates (D.20), (D.22) follow as in the previous lemma. The case $k = 0$ can be handled using interpolation and the estimates (D.4). When $k\ge 1$, we have

$$\begin{aligned}&\partial ^{\ell }\! D_t^k{\mathcal {F}} =\sum _{l_1+\cdots + l_n= k, \, n\ge 2} C_{l_1\cdots l_n}^k\partial ^\ell \big ((\widetilde{\partial }_{j_1} D_t^{l_1} S_{\epsilon }V^{j_2})\cdots (\widetilde{\partial }_{j_{n-1}} D_t^{l_{n-1}} S_{\epsilon } V^{j_n}) (\widetilde{\partial }_{j_n} D_t^{l_n}V^{j_1})\big )\nonumber \\&\quad =\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \sum _{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,l_1+\cdots + l_n= k,\,\, \sum |\beta _j|+|\gamma _j|=\ell -1\!\!\!\!\!\!\!\!\!} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \widetilde{C}_{l_{\!1\!}\cdots l_n}^k(\partial ^{\beta _{1}}\!\! A^{a_1}_{\,\,j_1}\!)\cdots (\partial ^{\beta _{n}{}_{\!}}\! A^{a_n}_{\,\,j_n}) (\partial _{a_1} \partial ^{\gamma _1} \!D_t^{l_1\!} S_\varepsilon V^{j_2}\!)\cdots \nonumber \\&\quad (\partial _{a_{{}_{\!}j_{n\!-\!1}}\!} \partial ^{\gamma _{n\!-\!1\!}}D_t^{l_{n\!-\!1\!}} S_\varepsilon V^{\!j_n\!})(\partial _{a_n} \partial ^{\gamma _n{}_{\!}}D_t^{l_n} V^{j_1}\!), \end{aligned}$$

(D.23)

where we have used (D.2) repeatedly. The leading order term is of the form

$$\begin{aligned} A_{\,\,i}^a (\partial ^\alpha D_t^k \partial _a S_\varepsilon V^j)(\widetilde{\partial }_j V^i)+(\partial ^\alpha A_{\,\,i}^a)(D_t^k\partial _aS_\varepsilon V^j) (\widetilde{\partial }_j V^j). \end{aligned}$$

We bound the first term by

$$\begin{aligned} C(M')||\partial V||_{L^\infty } ||D_t^k V||_{H^\ell (\Omega )}, \end{aligned}$$

and we bound the second term by

$$\begin{aligned}&P(||V(t)||_{\mathcal {X}^{s}}, ||\widetilde{x}(t)||_{H^{r-1}(\Omega )}), \quad \text {when}\,\, k\ge 1,\\&\quad \text {and}\quad C(M')||\partial V||_{L^\infty } ||\widetilde{x}||_{H^r(\Omega )},\quad \text {when}\,\, k=0. \end{aligned}$$

The lower order terms in (D.23) is controlled via Sobolev embedding. $\quad \square$

Writing $F_J\!\!=\!\!-(\widetilde{\partial }_i S_\varepsilon V_J^\ell )(\widetilde{\partial }_\ell V_J^i)$ for $J\!\! =\!\! I_{\!},{}_{\!}I{\!}I_{\!}$, a simple modification of the proof of Lemma D.10 gives

Lemma D.11

Suppose that (5.2) holds and let $s=k + \ell$. Then there is a continuous, positive function $C_s = C_s(M, ||V_I||_s,||V_{{}_{\!}I{\!}I}||_s, ||\,\widetilde{\!x}_{{}_{\!}I}||_{H^{s+1}}, ||\,\widetilde{\!x}_{{}_{\!}I\!I}||_{H^{s+1}})$ so that

$$\begin{aligned}&||D_t^k \big (F^1_{\!I\!} -\! F^1_{\!I\!I}\big )||_{H^{\ell }} \le C_s \big ( ||D_t^k V_{\!I} - D_t^k V_{\!I\!I}||_{H^{\ell +1}} \nonumber \\&\quad + ||\,\widetilde{\!x}_{{}_{\!}I}\!-\! \,\widetilde{\!x}_{{}_{\!}I\!I}||_{H^{\ell +1}} + ||V_{\!I} - V_{\!I\!I}||_{s} + ||\,\widetilde{\!x}_{{}_{\!}I}\!- \! \,\widetilde{\!x}_{{}_{\!}I\!I}||_{C^4_{x,t}(\Omega )}\big ),\qquad \end{aligned}$$

(D.24)

$$\begin{aligned}&||F^2(h_I) - F^2(h_{{}_{\!}I{\!}I})||_{s,0} \le C_s \big (||h_I - h_{II}||_{s+1, 0} + ||h_I - h_{II}||_s + ||h_I - h_{II}||_{C^3_{x,t}} \big ),\nonumber \\ \end{aligned}$$

(D.25)

$$\begin{aligned}&||F^2(h_I) - F^2(h_{{}_{\!}I{\!}I})||_{s-1} \le C_s \big (||h_I - h_{II}||_{s} + ||h_I - h_{II}||_{C^3_{x,t}} \big ). \end{aligned}$$

(D.26)

Appendix E: Existence of a Sequence of Compatible Data for the Smoothed Problem

In this section, our goal is to prove

Theorem E.1

Suppose that $V_0, h_0\!\in \! H^r\!\!$, $x_0\! \in \! H^{r+1}\!$ satisfy the compatibility conditions for Euler’s equations (2.15) to order $r\!-\!1 \!\ge \! 7$. Then there is a sequence of data $V_0^\varepsilon \!,h_0^\varepsilon \! \in \!H^r\!\!, \, x_0^\varepsilon \!\in \! H^{r+1}\!\!$ satisfying the compatibility conditions for the smoothed Euler’s equations (4.12) to order $r{}_{\!}-{}_{\!}1$, and $(V_0^\varepsilon \!, h_0^\varepsilon , x_0^\varepsilon )\! \rightarrow \! (V_0, h_0, x_0)$ as $\varepsilon \!\rightarrow \! 0$.

In the next section, we prove that if the compatibility conditions to order $r-1$ hold, given sufficiently regular V, the wave equation (2.9) has a solution $h(t,\cdot ) \in H^r(\Omega )$ with $D_t h(t,\cdot ) \in H^{r-1}(\Omega ),..., D_t^{r-1} h(t,\cdot ) \in H^1_0(\Omega ), D_t^r h(t,\cdot ) \in L^2(\Omega )$ for $t > 0$. We modify the approach of Lindblad–Luo [18] to construct functions $u_{-\!1}^\varepsilon , u_0^\varepsilon$ so that with $V_0^\varepsilon \!\!=\! V_0\! +\! \partial _{x_0} u_{-\!1}^\varepsilon$, $h_0^\varepsilon \! =\! h_0 \!+\! u_0^\varepsilon$, and $x_0^\varepsilon = x_0$, the initial data $V_0^\varepsilon \!, h_0^\varepsilon , x_0^\varepsilon$ satisfy the compatibility conditions (4.12). It will be convenient to reformulate the conditions used in Sects. 2.3 and 4.3 in a slightly more explicit way. Suppose that $\hat{x} \!=\! \!\sum x_k t^k\!/k!$, $\hat{V} \!\!=\! \!\sum V_k t^k\!/k!$, $\hat{h} \!=\! \!\sum V_k t^k\!/k!$ are formal power series solutions at $t \!=\! 0$ to (1.2)–(1.1) with $\hat{x}|_{t = 0} \!=\! x_0$ and $D_t^{\ell +1} \hat{x}|_{t = 0} \!=\! D_t^\ell \hat{V}|_{t = 0}$ and $\hat{x}_\varepsilon \!=\! \sum x_k^\varepsilon t^k\!/k!$, $\hat{V}_\varepsilon \!=\! \sum V_k^\varepsilon t^k\! / k!$, $\hat{h}_\varepsilon \!=\! \sum h_k^\varepsilon t^k\!/k!$ are power series solutions at $t\! = \!0$ to the smoothed problem (4.9)–(4.10) with $\hat{x}|_{t = 0\!} \!=\! x_0$ and $D_t^{\ell \!+\!1\!} \hat{x}^\varepsilon |_{t = 0\!} \!=\! D_t^\ell \hat{V}^\varepsilon |_{t = 0}$. Define

$$\begin{aligned}&F_k = \big ([D_t^k, \hat{\Delta }] \hat{h} + D_t^k (\hat{\partial }_i \hat{V}^j)(\hat{\partial }_j \hat{V}^i)\big )\big |_{t = 0},\\&\qquad G_k = \big (D_t^{k+1} (e'(\hat{h})D_t \hat{h}) -e'(\hat{h}) D_t^{k+2}\hat{h} + D_t^k\rho [\hat{h}]\big )\big |_{t = 0},\\&F^\varepsilon _k\! = \!\big ([D_t^k, \hat{\widetilde{\Delta }}] \hat{h}_\varepsilon \!+\! D_t^k (\hat{\widetilde{\partial }}_i S_\varepsilon \hat{V}_\varepsilon ^j)(\hat{\widetilde{\partial }}_j \hat{V}_\varepsilon ^i)\big )\big |_{t = 0},\\&\qquad G^\varepsilon _k \!=\! \big (D_t^{k+1} (e'(\hat{h}_\varepsilon )D_t \hat{h}_\varepsilon )\! -e'(\hat{h}_\varepsilon ) D_t^{k+2}\hat{h}_\varepsilon + D_t^k\rho [\hat{h}_\varepsilon ]\big )\big |_{t = 0}, \end{aligned}$$

as well as

$$\begin{aligned} C_k = [D_t^k, \hat{\partial }] (\hat{h} + \phi [\hat{x},\hat{h}]) |_{t = 0}, \qquad C_k^\varepsilon = [D_t^k,\hat{\widetilde{\partial }}] (\hat{h}_\varepsilon + \phi [\hat{x}_\varepsilon , \hat{h}_\varepsilon ]|_{t = 0}. \end{aligned}$$

Here, we are writing

$$\begin{aligned} \hat{\partial }_i = \frac{\partial {y}^a}{\partial \hat{x}^i} \frac{\partial }{\partial y^a} ,\qquad \hat{\widetilde{\partial }}_i = \frac{\partial {y}^a}{\partial \hat{\widetilde{x}}{}^i} \frac{\partial }{\partial y^a}, \qquad \hat{\Delta } = {\sum }_{i = 1}^3 \hat{\partial }^2_i, \qquad \hat{\widetilde{\Delta }} = {\sum }_{i = 1}^3 \hat{\widetilde{\partial }}_i{\!\!}^2. \end{aligned}$$

We are also writing $\phi [x, h]$ for the map $x, h \mapsto \phi$ defined in (2.5).

Taking the divergence of Euler’s equation (1.1) at $t = 0$ and subtracting it from the continuity equation (1.2) at $t = 0$ and performing the same manipulations to (4.9) and (4.10) gives that the coefficients $x_k, V_k, h_k, x_k^\varepsilon , V_k^\varepsilon , h_k^\varepsilon$ must satisfy the relations

$$\begin{aligned}&x_k = V_{k-1}, \qquad V_k = -\partial _{x_0} H_{k-1} +C_k, \qquad e'(h_0) h_{k+2} = \Delta h_k + F_k + G_k, \end{aligned}$$

(E.1)

$$\begin{aligned}&x^\varepsilon _k = V_{k-1}^\varepsilon , \qquad V_k^\varepsilon = -\partial _{x_0} H_{k-1}^\varepsilon + C_k^\varepsilon , \qquad e'(h_0^\varepsilon ) h_{k+2}^\varepsilon = \Delta h_k^\varepsilon + F_k^\varepsilon + G_k^\varepsilon , \end{aligned}$$

(E.2)

for $k \!\ge \!1$, with $H_{k-\!1} \!=\! h_{k-\!1} \!+ \phi _{k-\!1}, H_{k-\!1}^\varepsilon \! = \! h_{k-\!1}^\varepsilon \!+ \phi _{k-\!1}^\varepsilon$, and where $\phi _\ell \! =\! D_t^\ell \phi [\hat{x},\hat{h}]|_{t = 0}, \phi _\ell ^\varepsilon \!=\! D_t^\ell \phi [\hat{x}_\varepsilon , \hat{h}_\varepsilon ]$.

Expanding out the various definitions and replacing $x_k$ with $V_{k-1}$ for $k\ge 1$, it follows that

$$\begin{aligned} F_k= & {} F_k[x_0, V_0,..., V_k, h_0, ...., h_{k-1}], \qquad G_k = G_k[h_0,..., h_{k+1}], \nonumber \\ \phi _k= & {} K_k[x_0, V_0,..., V_{k}, h_0,...., h_k], \qquad C_k = C_k[x_0, V_0,..., V_{k-1}, H_0,..., H_{k-1}]. \nonumber \\ \end{aligned}$$

(E.3)

for functionals $K_k$ and where these functionals depend on space derivatives of their arguments, and similarly,

$$\begin{aligned} \begin{aligned} F_k^\varepsilon&= F_k^\varepsilon [x_0, V_0^\varepsilon ,..., V_k^\varepsilon , h_0^\varepsilon , ...., h_{k-1}^\varepsilon ],&\qquad G_k^\varepsilon&= G_k^\varepsilon [h_0^\varepsilon ,..., h_{k+1}^\varepsilon ], \\ \phi _k^\varepsilon&= K_k^\varepsilon [x_0, V_0^\varepsilon ,..., V_{k}^\varepsilon , h_0^\varepsilon ,...., h_k^\varepsilon ],&\qquad C_k^\varepsilon&= C_k^\varepsilon [x_0, V_0^\varepsilon ,..., V_{k-1}^\varepsilon , H_0^\varepsilon ,..., H_{k-1}^\varepsilon ]. \end{aligned} \end{aligned}$$

The formulas (E.3) combined with the second identity in (E.1) shows that $V_k$ can be expressed entirely in terms of $x_0, V_0$ and $h_0,..., h_{k-1}$ and similarly $V_k^\varepsilon$ can be expressed entirely in terms of $x_0, V_0, h_0^\varepsilon ,..., h_{k-1}^\varepsilon$. Consequently we will eliminate $V_k, V_k^\varepsilon$ for $k \ge 1$ from our equations and abuse notation slightly and write

$$\begin{aligned} F_k = F_k[x_0, V_0, h_0,..., h_{k-1}], \qquad F_k^\varepsilon = F_k^\varepsilon [x_0, V_0^\varepsilon , h_0^\varepsilon ,..., h_{k-1}^\varepsilon ]. \end{aligned}$$

E.0.1. The perturbative system. We start by considering the following system, with $\Delta = \sum _{i = 1}^3 \partial _{i}^2$:

$$\begin{aligned}&\Delta u_{-1}^\varepsilon = -e'(h_0 + u_0^\varepsilon ) u_1^\varepsilon ,&\text { in } \Omega ,&\nonumber \\&\Delta u_k^\varepsilon = F_k - \widetilde{F}_k^\varepsilon + G_k - \widetilde{G}_k^\varepsilon + e'(h_0 + u_0^\varepsilon )u_{k+2}^\varepsilon ,&\text { in } \Omega , \text { for } k = 0,..., r-2, \end{aligned}$$

(E.4)

$$\begin{aligned}&u_k^\varepsilon = 0,&\text { on } \partial \Omega , \text { for } k = -1,..., r-2, \end{aligned}$$

(E.5)

with $\widetilde{F}_k^\varepsilon = \widetilde{F}_k^\varepsilon [u_{-1}^\varepsilon ,..., u_k^\varepsilon ] ,\widetilde{G}_k^\varepsilon = \widetilde{G}_k^\varepsilon [u_0^\varepsilon ,..., u_{k+1}^\varepsilon ]$ defined by

$$\begin{aligned}&\widetilde{F}_k^\varepsilon = F_k^\varepsilon [x_0, V_0 + \widetilde{\partial }u_{-1}^\varepsilon , h_0 + u_0^\varepsilon ,..., h_{k-1} + u_{k-1}^\varepsilon ],\\&\quad \widetilde{G}_k^\varepsilon = G_k^\varepsilon [h_0 + u_0^\varepsilon ,...., h_{k+1} + u_{k+1}^\varepsilon ], \end{aligned}$$

and with the convention that $u_\ell ^\varepsilon = 0$ for $\ell \ge r-1$.

Suppose for the moment that this system has a solution $(u_{-1}^\varepsilon ,..., u_{r-1}^\varepsilon )$. We claim that with $V_0^\varepsilon = V_0 + \partial _{x_0} u_{-1}^\varepsilon$ and $h_0^\varepsilon = h_0 + u_0^\varepsilon$, the initial data $(V_0^\varepsilon , h_0^\varepsilon )$ satisfy the compatibility conditions (4.16) for the smoothed problem to order $r-1$. Indeed, because $h_0 = 0$ on $\partial \Omega$ and because of the boundary condition (E.5) we have that $h_0^\varepsilon = 0$ on $\partial \Omega$. To see that $h_1^\varepsilon = 0$ on $\partial \Omega$, we note that by construction,

$$\begin{aligned} e'(h_0^\varepsilon )h_1^\varepsilon = - {{\,\mathrm{div}\,}}V_0^\varepsilon = - {{\,\mathrm{div}\,}}V_0 - \Delta u_{-1}^\varepsilon = e'(h_0)h_1 + e'(h_0^\varepsilon )u_1^\varepsilon . \end{aligned}$$

By the compatibilty conditions for $V_0, h_0$, we have $h_1 \!=\! 0$ on $\partial \Omega$ and by construction $u_1^\varepsilon \!=\! 0$ on $\partial \Omega$ and so the first compatibility condition (4.12) holds as well. Using the definitions of $h_2^\varepsilon , h_2$ from (E.1), (E.2), we have

$$\begin{aligned} e'(h_0^\varepsilon ) h_2^\varepsilon = \Delta h_0 + \Delta u_0^\varepsilon + F_0^\varepsilon + G_0^\varepsilon = \Delta h_0 + F_0 + G_0 + e'(h_0^\varepsilon )u_2^\varepsilon = e'(h_0)h_2 + e'(h_0^\varepsilon ) u_2^\varepsilon , \end{aligned}$$

By the compatibility conditions, $h_2 = 0$ on $\partial \Omega$ and this combined with the boundary condition (E.5) shows that $h_2^\varepsilon = 0$ on $\partial \Omega$ as well. In general, this construction gives that

$$\begin{aligned}&e'(h_0^\varepsilon ) h_k^\varepsilon = e'(h_0) h_k + e'(h_0^\varepsilon ) u_k^\varepsilon , \,\, k = 0,..., r-2, \\&\quad e'(h_0^\varepsilon ) h_k^\varepsilon = e'(h_0)h_k, \,\,\, k = r-1, r, \end{aligned}$$

from which it immediately follows that the compatibility condition of order $r-1$ holds for the smoothed problem so long as the compatibility condition of order $r-1$ holds for the original problem.

Because $e'$ is assumed to be small, a simplified model for the above system is the following:

$$\begin{aligned} \Delta w_{-1} = \kappa w_{1},\quad \Delta w_k = {\sum }_{\ell \le k-1} A_k^\ell w_\ell + f_k + \kappa w_{k+2}, \quad k = 0,..., N, \qquad \text { in } \Omega , \end{aligned}$$

(E.6)

with the boundary condition $w_k = 0$ on $\partial \Omega$ for all k. Here, $A_k^\ell , f_k$ are given functions, $\kappa$ is a small parameter and we are writing $w_\ell = 0$ for $\ell \ge N+1$. When $\kappa = 0$, this system is lower-triangular and can be solved directly by successively solving for $w_0, w_1,...$. To solve the model system (E.6) for nonzero but small $\kappa$, one can use the following iteration: $w^0_k \equiv 0$ for all k and then, given $w^{\nu -1}_k$, solve the following system for $w^\nu _k$:

$$\begin{aligned} \Delta w_{{-1}}^\nu = \kappa w_2^{\nu -1},\quad \Delta w_k^{\nu } = {\sum }_{\ell \le k-1} A_k^\ell w_\ell ^\nu + f_k + \kappa w_{k+2}^{\nu -1}, \qquad k = 0,..., N-1, \end{aligned}$$

with $w^\nu _k = 0$ on $\partial \Omega$. Writing $W_k^\nu = w_k^\nu - w_k^{\nu -1}$, by standard elliptic theory there are estimates of the form

$$\begin{aligned} ||W_k^{\nu }||_{H^{s-k}} \le C\big ({\sum }_{\ell \le k-1} ||W_\ell ^{\nu }||_{H^{s-k-2}} + \kappa ||W^{\nu -1}||_{H^{s-k-2}}\big ),\qquad k =-1...., N, \end{aligned}$$

where C depends on norms of the coefficients A. Iterating this estimate leads to an inequality of the form

$$\begin{aligned} ||W_k^\nu ||_{s-k} \le {\sum }_{\mu = 1}^{\nu -1} (C\kappa )^\mu . \end{aligned}$$

For $\kappa$ sufficiently small, the sequence $w^\nu \!$ converges as $\nu \!\rightarrow \!\infty$ to a solution $w \!= \!(w_{-1},\dots ,w_{N})$ satisfying (E.6).

E.0.2. The iteration to solve the system. In order to solve the system (E.4)–(E.5), we will use the following iteration. We set $u_k^0 \equiv 0$ in $\Omega$ for $k = -1,..., r$ and for $\nu \ge 1$, we define $u_{-1}^\nu ,...,u_r^\nu$ by $u_{r-1}^\nu = u_r^\nu = 0$ and

$$\begin{aligned}&\Delta u_{-1}^\nu = -e'(h_0 + u_0^{\nu -1}) u_1^{\nu -1},&\text { in } \Omega ,&\end{aligned}$$

(E.7)

$$\begin{aligned}&\Delta u_k^\nu = F_k - F_k^{\nu } + G_k-G_k^{\nu -1}+e'(h_0 + u_0^{\nu -1}) u_{k+2}^{\nu -1},&\text { in } \Omega , \text { for } k = 0,..., r-2, \end{aligned}$$

(E.8)

$$\begin{aligned}&u_k^\nu = 0,&\text { on } \partial \Omega , \text { for } k = -1,..., r-2, \end{aligned}$$

(E.9)

where we are writing

$$\begin{aligned} F_k^{\nu } = \widetilde{F}_k^\varepsilon [u_{-1}^\nu ,..., u_{k-1}^\nu ], \qquad G_k^{\nu -1} = \widetilde{G}_k^\varepsilon [u_0^{\nu -1},..., u_{k+1}^{\nu -1}]. \end{aligned}$$

Let $u^\nu = (u_{-1}^\nu ,...,u_r^\nu )$. To see that this system has a solution $u^\nu$ given $u^{\nu -1}$, one just uses the fact that it is lower-triangular in $u^\nu$; first solve (E.7) for $u_{-1}^\nu$ and then solve (E.8)–(E.9) successively for $k = 0, 1,...,r-2$.

We will prove that the sequence $u^\nu$ is uniformly bounded in $\nu$ in the norm

$$\begin{aligned} ||u^\nu ||_r = || \partial _{x_0} u_{-1}^\nu ||_{H^r(\Omega )} + ||u_{-1}^\nu ||_{H^r(\Omega )} + {\sum }_{k = 0}^{r-2} ||u_k^\nu ||_{H^{r-k}(\Omega )}. \end{aligned}$$

Set $E_0 = ||V_0||_{H^r}^2 + ||h_0||_{H^r}^2 + ||x_0||_{H^{r+1}}^2$. In the following sections we will prove

Proposition E.2

Fix $r\! \ge \! 7$. There is a continuous function $C_r$ so that if $u^\nu$ satisfies (E.7)–(E.9), then

$$\begin{aligned} ||u^\nu ||_r \le C_r(E_0, ||u^{\nu -1}||_{r-1}) (\kappa ||u^{\nu -1}||_r + \varepsilon ), \end{aligned}$$

(E.10)

and there is a continuous function $D_r$ so that:

$$\begin{aligned} ||u^\nu - u^{\nu -1}||_{r} \le D_r(E_0, ||u^\nu ||_{r}, ||u^{\nu -1}||_{r}) \kappa ||u^{\nu -1} - u^{\nu -2}||_r. \end{aligned}$$

(E.11)

Let us now explain why one should expect estimates of this form. The estimates (E.10) follow from elliptic estimates applied to the system (E.7)–(E.9) and will ultimately follow from estimates for $F_k - F_k^\nu , G_k - G_k^{\nu -1}$ in Sobolev spaces. Let us consider the $k = 0$ case. Using that $\hat{x}|_{t = 0} = x_0$ we have

$$\begin{aligned} F_0 - F_0^{\nu } = (\partial _i V_0^j)(\partial _j V_0^i) - (\partial _i S_\varepsilon (V_0^j + \delta ^{jk}\partial _k u_{-1}^\nu ) (\partial _j (V_0^i+\delta ^{ik}\partial _k u_{-1}^\nu )). \end{aligned}$$

Expanding this out generates several terms, but let us just consider two of them:

$$\begin{aligned} \partial _i V_0^j (\delta ^{ij}\partial _j \partial _k u^{\nu }_{-1}) \quad \text { and } \quad (\partial _i V_0^j - \partial _i S_\varepsilon V_0^j)(\partial _j V_0^i). \end{aligned}$$

To control the $L^2(\Omega )$ norm, say, of the first term we use the equation (E.7) and standard elliptic theory to control $||u^{\nu }_{-1}||_{H^2(\Omega )} \le C ||e'(u_0^{\nu -1}) u_1^{\nu -1}||_{L^2(\Omega )}$. With $\kappa \ge \sup |e'|$ this type of term can be bounded by the first term in (E.10). Also, we have $||V_0 - S_\varepsilon V_0||_{H^1(\Omega )} \le C \varepsilon ||V_0||_{H^2(\Omega )}$ so the second type of term can be bounded by the second term in (E.10). Assuming that (E.10)–(E.11) hold for the moment, we give the proof as follows

Proof of Theorem E.1

With the function $C_r$ from Proposition E.2, take $C_0 = \max _{z \in [0,1]} C_r(E_0, z)$. Also take $\kappa$ so small that $2\kappa C_0 \le 1$ and $\varepsilon$ so small that $2\varepsilon {\sum }_{\mu = 0}^\infty (\kappa C_0)^\mu \le 1$. Since $u^{0} =0$, it follows from (E.10) that $||u^1||_r \le C_0 \varepsilon$. By induction it then follows that $||u^\nu ||_{r} \le \varepsilon \sum _{\mu = 0}^\nu (\kappa C_0)^\mu$, and by the assumption on $\kappa$ the sum on the right-hand side is uniformly bounded as $\nu \rightarrow \infty$.

Next, with the function $D_r$ from Proposition E.2, take $D_0 = \max _{z_1,z_2 \in [0,1]} D_r(E_0,z_1,z_2)$. By induction and (E.11) it follows that $||u^\nu - u^{\nu -1}||_r \le \varepsilon (\kappa D_0)^{\nu }$. Therefore, $u^\nu$ is a Cauchy sequence and so it converges to some limit u which satisfies the perturbative system (E.4)–(E.5) by construction. To prove the second point in the theorem, taking $\nu \!\rightarrow \! \infty$ in the estimate for $u^\nu$ that we just proved shows $||u||_r \le \varepsilon \sum _{\mu = 0}^\infty (\kappa C_0)^\mu \le \varepsilon$. $\quad \square$

We will use the following estimate, which is a straightforward consequence of the elliptic estimate (5.8) at $t = 0$: If $s \ge 2$, there is a constant $C_s = C_s(||x_0||_{H^{s+1}})$ so that if $f = 0$ on $\partial \Omega$, then

$$\begin{aligned} ||\partial _{x_0} f||_{H^{s}} \le C_s ||\Delta _{x_0} f||_{H^{s-1}}, \qquad ||f||_{H^s} \le C_s ||\Delta _{x_0}f||_{H^{s-2}}. \end{aligned}$$

(E.12)

The estimates (E.10) and (E.11) follow after repeatedly applying the next lemma.

Lemma E.3

There are continuous functions $C_{r,k}$ so that if $u^\nu = (u_{-1}^\nu ,..., u_{r-2}^{\nu })$ satisfies the approximate system (E.7)–(E.9) and if the equation of state satisfies (2.11), then

$$\begin{aligned} ||u_{-1}^\nu ||_{H^r}&\le C_{r,-1}(E_0, ||u^{\nu -1}_0||_{H^{r-2}}) \kappa ||u_1^{\nu -1}||_{H^{r-2}}, \end{aligned}$$

(E.13)

$$\begin{aligned} ||u_k^\nu ||_{H^{r-k}}&\le C_{r,k}(E_0, ||u^{\nu -1}||_r, {{\,\mathrm{{\textstyle {\sum }}}\,}}_{\ell \le k-1} ||u^{\nu }_\ell ||_{H^{r-\ell -1}}) \big ( {\sum }_{\ell \le k-1} ||u_\ell ^\nu ||_{H^{r-k}} \nonumber \\&\quad + \kappa ||u^{\nu -1}||_r + \varepsilon \big ), \end{aligned}$$

(E.14)

and there are continuous functions $D_{r,k}$ so that with $U_k^\nu = u_k^\nu - u_k^{\nu -1}$,

$$\begin{aligned} ||U_{-1}^\nu ||_{H^r}&\le D_{r,-1}(E_0, ||u_0^{\nu -1}||_r, ||u_0^{\nu -2}||_{r}) \kappa ||U^{\nu -1}_1||_{H^{r-2}}, \end{aligned}$$

(E.15)

$$\begin{aligned} ||U_{k}^\nu ||_{H^{r-k}}&\le D_{r,k}(E_0, ||u^{\nu }||_r, ||u^{\nu -1}||_{r}, ||u^{\nu -2}||_r) \big ( {\sum }_{\ell \le k-1} ||U_\ell ^{\nu }||_{H^{r-\ell }} \nonumber \\&\quad + \kappa ||U_\ell ^{\nu -1}||_{H^{r-\ell }}\big ). \end{aligned}$$

(E.16)

Proof

Using the elliptic estimates (E.12) and the fact that $H^{r-k-2}(\Omega )$ is an algebra for $r -k \ge 4$, we have

$$\begin{aligned} ||\partial _{x^0}u^\nu _{-1}||_{H^r}&\le C \big ( ||e'(h_0 + u_0^{\nu -1})||_{H^{r-1}} ||u_1^{\nu -1}||_{H^{r-1}}\big ),\\ ||u^\nu _k||_{H^{r-k}}&\le C \big (||F_k - F_k^\nu ||_{H^{r-k-2}} + ||G_k - G_k^{\nu -1}||_{H^{r-k-2}}\\&\quad + ||e'(h_0 + u_0^{\nu -1})||_{H^{r-k-2}} ||u_{k+2}^{\nu -1}||_{H^{r-k-2}}\big ), \end{aligned}$$

for $k = 0,..., r-2$, with the convention that $u_\ell ^\nu = 0$ for $\ell \ge r-1$, and with constants depending on $||x_0||_{H^{r+1}}$.

Because $U^\nu = u^\nu - u^{\nu -1}$ satisfies the following system in $\Omega$:

$$\begin{aligned} \Delta U_{-1}^\nu&= e'(h_0 + u_0^{\nu -1}) u_1^{\nu -1} - e'(h_0 + u_0^{\nu -2}) u_1^{\nu -2},\\ \Delta U_k^\nu&= F^{\nu -1}_k - F_k^\nu + G^{\nu -1}_k - G^\nu _k\\&\quad +e'(h_0 + u_0^{\nu -1}) u_{k+2}^{\nu -1} - e'(h_0 + u_{0}^{\nu -2}) u_{k+2}^{\nu -2}, \quad k = 0,..., r-2, \end{aligned}$$

with $U^\nu = 0$ on $\partial \Omega$, we also have

$$\begin{aligned}&||\partial _{x_0}U_{-1}^\nu ||_{H^r} \le C \big ( ||e'(h_0 + u_0^{\nu -1}) - e'(h_0 + u_0^{\nu -2})||_{H^{r-1}} ||u_1^{\nu -1}||_{H^{r-1}}\\&\quad + ||e'(h_0 + u_0^{\nu -2})||_{H^{r-1}} ||U_1^{\nu -1}||_{H^{r-1}}\big ),\\&\quad ||U_k^\nu ||_{H^{r-k}} \le C\big ( ||F^{\nu -1}_k - F^{\nu }_k||_{H^{r-k-2}} + ||G^{\nu -1}_k - G^{\nu }_k||_{H^{r-k-2}}\\&\quad +||e'(h_0 + u_0^{\nu -1}) - e'(h_0 + u_0^{\nu -2})||_{H^{s-k-2}} ||u_{k+1}^{\nu -1}||_{H^{r-k-2}} \\&\quad + ||e'(h_0 + u_0^{\nu -2})||_{H^{r-k-2}} ||U_{k+1}^{\nu -1}||_{H^{r-k-2}}. \end{aligned}$$

The estimates (E.13)–(E.16) then follow from Proposition E.4 and Lemmas E.5–E.6. $\quad \square$

It remains to prove estimates for the terms on the right-hand sides of (E.13), (E.14) and (E.15), (E.16). The proposition below is a consequence of Lemmas E.8, E.9, whose proofs we postpone until Section E.1

Proposition E.4

Set $M_k^\nu = ||\partial _{x_0}u_{-1}^\nu ||_{H^{r}} + {{\,\mathrm{{\textstyle {\sum }}}\,}}_{j \le k} ||u_j^\nu ||_{H^{r-j}}$. There are continuous functions $K_k = K_{k}(E_0, M_{k-1}^\nu ), K_k' = K'_{k}(E_0, M_{k-1}^\nu , M_{k-1}^{\nu -1})$ so that writing $j = r-k-2$,

$$\begin{aligned}&||{}_{\!}F_{\!k} - F_{\!k}^{\nu }||_{{}_{\!}H^j}\! \!\le \! K_{\!k}\big ( ||u_{-\!1}^\nu \!||_{{}_{\!}H^{j\!+2}} + \!{\sum }_{\ell \le k-\!1\!} ||u_{\ell }^\nu ||_{{}_{\!}H^{j\!+2}} +\varepsilon \big ),\\&\qquad ||{}_{\!}F_{\!k}^\nu \!\!- F_{\!k}^{\nu \!-\!1\!}||_{{}_{\!}H^j} \!\!\le \! K_{\!k}' \big ( ||U_{\!-\!1\!}^\nu ||_{{}_{\!}H^{j\!+2}} + \!{\sum }_{\ell \le k-\!1\!} ||U_{\ell }^\nu ||_{{}_{\!}H^{j\!+\!2}}\!\big ). \end{aligned}$$

Lemma E.5

There are continuous functions $K= K(E_0, ||u^{\nu -1}||_{r-1})$, $K' = K'(E_0,||u^{\nu -1}||_r, ||u^{\nu -2}||_{r})$ so that if $\sup _{r' \le r+1} |e^{(r')}| \le \kappa$ then

$$\begin{aligned} ||G_k - G_k^{\nu -1}||_{H^{r-k-2}} \le \kappa K||u^{\nu -1}||_{r}, \qquad ||G_k^{\nu -1} - G_k^{\nu -2}||_{H^{r-k-2}} \le \kappa K' ||u^{\nu -1} - u^{\nu -2}||_r \end{aligned}$$

Proof

Write $h_k^\nu = h_k + u_k^{\nu -1}$. Expanding out the definition of $G_k, G_k^{\nu }$ and applying $\partial _y^I$ for a multi-index I with $|I| = r' \le r-k - 2$, we see that $\partial _y^I (G_k - G_k^{\nu })$ is a sum of terms of the form

$$\begin{aligned} e^{(K)}(h_0^{\nu -1}) (\partial _y^{J_1} h_{k_1}^{\nu -1}) \cdots (\partial _y^{J_j} h_{k_j}^{\nu -1}) - e^{(K)}(h_0) (\partial _y^{J_1} h_{k_1}) \cdots (\partial _y^{J_j} h_{k_j}), \end{aligned}$$

with $|J_1| + \cdots |J_j| = r-k-2, k_1 + \cdots k_j = k+1, K \le r-1$. Performing the usual manipulations, rearranging terms, and using that $h_k^{\nu -1} \!\!- h_k = u_k^{\nu -1}\!\!$, it suffices to control the $L^2(\Omega )$ of a sum of terms of the forms

$$\begin{aligned}&(e^{(K)}(h_0^{\nu -1}) - e^{(K)}(h_0)) (\partial _y^{J_1}h^{\nu -1}_{k_1}) \cdots (\partial _y^{J_j}h^{\nu -1}_{k_j}),\\&\quad \text { and } \quad e^{(K)}(h_0^{\nu -1}) (\partial _y^{J_1}u_{k_1}^{\nu -1}) (\partial _y^{J_2}h^{\nu -1}_{k_2}) \cdots (\partial _y^{J_j}h^{\nu -1}_{k_j}), \end{aligned}$$

the remaining terms being similar but with some of the factors of $h^{\nu -1}_\ell$ replaced by $h_\ell$. Let us just bound the second type of term here, the first type being identical after using the estimate $|e^{(K)}(h_0^{\nu -1}) - e^{(K)}(h_0)| \le |\sup e^{(K+1)}| |u_0^{\nu -1}|$. For each $\ell$ with $|J_\ell | + k_\ell \le r-3$, we bound the resulting term in $L^\infty$ by Sobolev embedding to get either $||\partial _y^{J_\ell } u^{\nu -1}_{k_\ell }||_{L^\infty (\Omega )} \le C|| \partial _y^{J_\ell }u^{\nu -1}_{k_\ell }||_{H^2(\Omega )}$ or $C(||\partial _y^{J_\ell } h_{k_\ell }||_{H^2(\Omega )} + ||\partial _y^{J_\ell } u_{k_\ell }^{\nu -1}||_{H^2(\Omega )})$. Since $|J_\ell |+ k_\ell + 2 \le r-1$, the result can be bounded by $||u^{\nu -1}||_{s-1}$ or $||u^{\nu -1}||_{r-1} + E_0$, respectively. It therefore remains to handle terms with $|J_\ell | + k_\ell \ge r-2$. Since $r \ge 5$ there is at most one such term and so it is bounded by either $||u_{k_\ell }^{\nu \!-\!1}||_{H^{|\!J_\ell \!|}(\Omega )} \le ||u^{\nu \!-\!1}||_{r-\!1}$ or $||h_{k_\ell }||_{H^{|\!J_\ell \!|}(\Omega )} + ||u_{k_\ell }^{\nu \!-\!1}||_{H^{|\!J_\ell \!|}(\Omega )} \le E_0 + ||u^{\nu \!-\!1}||_{r-\!1}$, as required. The estimate for $G^{\nu -1} - G^{\nu -2}$ is similar. $\quad \square$

Lemma E.6

There are continuous functions $K'' \!\!= \!K''(E_0,\! ||u_0^{\nu -\!1\!}||_{H^{r\!-\!1}}\!)$, $K^{\prime \prime \prime } \!\!= \!K^{\prime \prime \prime }(E_0, ||u_0^{\nu -\!1\!}||_{H^{r\!-\!1}\!}, \!||u_0^{\nu -2\!}||_{H^{r\!-\!1}\!})$ so that if $\sup _{k \le r+1} |e^{(k)}| \le \kappa$ then,

$$\begin{aligned} ||e'(h_0^{\nu -1})||_{H^{r}} \le \kappa K'' ||u_0^{\nu -1}||_{H^r(\Omega )}, \qquad ||e'(h_0^{\nu -1}) - e'(h_0^{\nu -2})||_{H^r}\le \kappa K''' ||u_0^{\nu -1} - u_0^{\nu -2}||_{H^r}. \end{aligned}$$

(E.17)

Proof

By the chain rule, if $I\!$ is a multi-index with $|I|\! = \!r' \!\!\le \!r$, $\partial _y^I (e'(h_0 \!+\! u_0^{\nu -1}))\!$ is a sum of terms of the form

$$\begin{aligned} e^{(K)}(h_0 + u_0^{\nu -1}) (\partial _y^{J_1} h_0 + u_0^{\nu -1}) \cdots (\partial _y^{J_j} h_0 + u_0^{\nu -1}) , \qquad {{\,\mathrm{{\textstyle {\sum }}}\,}}|J_j| = r', K\le r'. \end{aligned}$$

We want to control the $L^2(\Omega )$ norm of this. For each $\ell$ with $|J_\ell |\! \le \!r\!-\! 3$ we control the $L^\infty$ norm of the resulting factor by Sobolev embedding which shows that any such term is bounded by $C (||h_0||_{H^{r-1}(\Omega )} + ||u_0^{\nu -1}||_{H^{r-1}(\Omega )})$. To handle terms with $|J_\ell | \!\ge \! r\!-\!2$, note that since $r\! \ge \! 5$ there can be at most one such term and we control it by $||u_0^{\nu -1}||_{H^r(\Omega )}$. Since $|e^{(K)}| \le \kappa$ this gives the first estimate (E.17) and the second is similar. $\quad \square$

1.1 E.1: Estimates for $F_k - F_k^\nu$ and $F_k^\nu - F_k^{\nu -1}$

For these estimates it will be convenient to first state the results in terms of the coefficients $V_k, V_k^\nu$ before relating these to $h_k,u_k^\nu$, because they depend on each other in a complicated way. Recall the definitions of $S, \tilde{S}$ from (2.13), (4.14). Given power series in time t$\hat{V}\!, \hat{V}_\varepsilon$ as in the beginning of this section and evaluating at $t \!= \!0$, the S, $\widetilde{S}$ are polynomials in the following arguments:

$$\begin{aligned} S_\ell ^k = S_\ell ^k(\partial V_0, ..., \partial V_{k-\ell -1}), \qquad \widetilde{S}_\ell ^k = \widetilde{S}_\ell ^k(\partial S_\varepsilon V_0^\varepsilon ,..., \partial S_\varepsilon V_{k-\ell -1}^\varepsilon ). \end{aligned}$$

We note for later use that in fact we have

$$\begin{aligned} S_\ell ^k(\partial V_0, ..., \partial V_{k-\ell -1})= \widetilde{S}_\ell ^k(\partial V_0,..., \partial V_{k-\ell -1}), \end{aligned}$$

(E.18)

which follows from the formulas (2.13), (4.14). We then have the following formula for the $V_k$:

$$\begin{aligned} V_k^i = -\delta ^{ii'}\partial _{i'} H_{k-1} + {\sum }_{\ell \le k-2} \delta ^{ii'}S_{i'\ell }^{jk} \partial _j H_\ell , \qquad i = 1,2,3, \end{aligned}$$

(E.19)

and, given $V_0^\nu$, we recursively define $V_k^\nu$ by

$$\begin{aligned} V_k^{i \nu } = -\delta ^{ii'}\partial _{i'} H_{k-1}^\nu + {\sum }_{\ell \le k-2} \delta ^{ii'}\widetilde{S}^{jk,\nu }_{i' \ell } \partial _j H_\ell ^\varepsilon , \qquad i = 1,2,3, \end{aligned}$$

(E.20)

where, with $\tilde{S}^{jk}_{i\ell }$ defined in (4.14), we are writing

$$\begin{aligned} \widetilde{S}^{j k,\nu }_{i\ell } = \widetilde{S}^{j k}_{i\ell }( \partial S_\varepsilon V_0^\nu ,..., \partial S_\varepsilon V_{k-\ell -1}^\nu ). \end{aligned}$$

We are also writing $H_\ell = h_\ell + \phi _\ell$ and $H_\ell ^\nu = h_\ell ^\nu + \phi _\ell ^\nu$ with $\phi _\ell ^\nu = D_t^\ell \phi [x,\hat{h}^\nu ]|_{t = 0}$, where $\hat{h}^\nu (t) = \sum h^\nu t^k\!/k!$.

If T is a (2,2) tensor then we write

$$\begin{aligned} ||T_\ell ^k||_{H^m}^2 = {\sum }_{0 \le |I| \le m} \int _\Omega \delta ^{ii'}\delta _{jj'} (\partial _y^I T_{i \ell }^{jk})(\partial _y^I T_{ i' \ell }^{j'k}) \mathrm{d}y, \end{aligned}$$

and we have the following lemma which will be used repeatedly to control the commutators $S, \widetilde{S}$:

Lemma E.7

Let $e_0^s = E_0 + \sum _{j \le s} ||V_j||_{H^{s-j}}$. If $s \ge 2$ then there are continuous functions $C_{k,\ell }$ so that

$$\begin{aligned}&||S_\ell ^k||_{H^{s}} \le C_{k,\ell }(e_0^{s+1}), \qquad ||\widetilde{S}_{\ell }^{k,\nu }||_{H^{s}} \le C_{k,\ell }(e_0^{s+1},m_{k-\ell +1,s+1}^{\nu }),\\&\quad \text {where}\quad m_{r,s+1}^\nu = {{\,\mathrm{{\textstyle {\sum }}}\,}}_{j \le r} ||V_j^\nu ||_{H^{s+1}}, \end{aligned}$$

and there are continuous functions $D_{k,\ell }, D_{k,\ell }'$ so that

$$\begin{aligned} ||S_\ell ^k - \widetilde{S}_\ell ^k||_{H^s}&\le D_{k,\ell }(e_0^{s+1}, m_{k-\ell +1,s+1}^\nu ) ({\sum }_{j \le k-\ell -1}||V_{j} - V_{j}^{\nu -1}||_{H^{m+1}} +\varepsilon e_0^{s+2}), \end{aligned}$$

(E.21)

$$\begin{aligned} ||\widetilde{S}_\ell ^{k,\nu } - \widetilde{S}_\ell ^{k,\nu -1}||_{H^{s}}&\le D_{k,\ell }'(e_0^{s+1}, m_{k-\ell +1,s+1}^\nu , m_{k-\ell +1,s+1}^{\nu -1}) {\sum }_{j \le k-\ell -1} ||V_j^{\nu } - V_{j}^{\nu -1}||_{H^{s+1}}. \end{aligned}$$

(E.22)

Proof

Because $s \ge 2, H^s(\Omega )$ is an algebra and so the first two estimates follow because $S, \widetilde{S}$ are polynomials in their arguments (see (2.13) and (4.14)).

To prove (E.21), set $A \!=\! (\partial V_0,..., \partial V_{k-\ell -1})$ and $\widetilde{A}^\nu \!=\! (\partial S_\varepsilon V_0^\nu ,..., \partial S_\varepsilon V_{k-\ell -1}^\nu )$. Abusing notation, we write

$$\begin{aligned} S_\ell ^k - \widetilde{S}^{k,\nu }_\ell = S^k_\ell (A) - \widetilde{S}_\ell ^k(\widetilde{A}^\nu ) = S_\ell ^k(A) - \widetilde{S}_\ell ^k(A) + (\widetilde{S}_\ell ^k(\widetilde{A}^\nu ) - \widetilde{S}^k_\ell (A)). \end{aligned}$$

By (E.18), the first two terms cancel. Since $\widetilde{S}$ is a polynomial in its arguments, we have

$$\begin{aligned} ||\widetilde{S}_\ell ^k(\widetilde{A}^\nu ) - \widetilde{S}_\ell ^k(A) ||_{H^s} \le C' {\sum }_{j \le k-\ell -1} ||V_j - S_\varepsilon V_j^\nu ||_{H^{s+1}}, \end{aligned}$$

with $C = C(E_0^s, ||A||_{H^{s}}, ||A^\nu ||_{H^{s}})$, after additionally using that $S_\varepsilon$ is bounded on Sobolev spaces. Now we write $||V_j - S_\varepsilon V_j^\nu ||_{H^{s+1}} \le ||V_j - S_\varepsilon V_j||_{H^{s+1}} + ||S_\varepsilon (V_j - V_j^\nu )||_{H^{s+1}}$. Since $||V_j - S_\varepsilon V_j||_{H^{s+1}} \le C \varepsilon ||V_j||_{H^{s+2}}$ by (A.12), this concludes the proof of the third estimate. The proof of (E.22) is similar. $\quad \square$

We have the following technical estimate for $F_k -F_k^\nu$ and $F_k^\nu - F_k^{\nu -1}$ in terms of $V_k, V_k^\nu$:

Lemma E.8

Set $m_k^\nu = ||V_0^\nu ||_{H^{r}} + {{\,\mathrm{{\textstyle {\sum }}}\,}}_{0 \le j \le k} ||V_j^\nu ||_{H^{r-j-1}}$. There are continuous functions $K = K_{r,k}(E_0, m_{k}^\nu ), K' = K'_{r,k}(E_0, m_k^\nu , m_k^{\nu -1})$ so that, with $V_k, V_k^\nu$ defined by (E.19)–(E.20) and $j = r-k-2$,

$$\begin{aligned}&||F_k - F_k^{\nu }||_{H^j} \le K\big ( ||u_{-1}^\nu ||_{H^{j+2}}+ {\sum }_{\ell \le k} ||V_\ell - V_\ell ^\nu ||_{H^{j+1}} + ||u^\nu _{\ell -1}||_{H^{j+2}} +\varepsilon \big ), \\&||F_k^\nu - F_k^{\nu -1}||_{H^j} \le K' \big ( ||u_{-1}^\nu - u_{-1}^{\nu -1}||_{H^{j+2}} \\&\quad + {\sum }_{\ell \le k} ||V_\ell ^\nu - V_\ell ^{\nu -1}||_{H^{j+1}} + ||u_\ell ^\nu - u_\ell ^{\nu -1}||_{H^{j+2}}\big ). \end{aligned}$$

Proof

We start by writing $F_k - F_k^\nu$ more explicitly in terms of the S and $\widetilde{S}$. With $V_k^\nu$ defined in (E.20) and with $h^\nu _k = h_k + u_k^\nu$, let $\hat{V}^\nu (t) = \sum _{k = 0}^N V_k^\nu t^k/k!$ and $\hat{h}^\nu (t) = \sum _{k = 0}^N h_k^\nu t^k/k!$, we write

$$\begin{aligned} F_k - F_k^\nu = D_t^k \big ( (\hat{\partial } \hat{V})(\hat{\partial } \hat{V}) - (\hat{\widetilde{\partial }} S_\varepsilon \hat{V}^\nu )(\hat{\widetilde{\partial }} \hat{V}^\nu ) \big )|_{t = 0} + \big ([D_t^k, \hat{\Delta }] \hat{h} - [D_t^k, \hat{\widetilde{\Delta }}] \hat{h}^\nu \big )|_{t = 0} \equiv f_k^\nu + g_k^\nu . \end{aligned}$$

For matrices $a_i^j, b_i^j$, write $a\cdot b = a_i^j b_j^i$ and if T is a (2,2) tensor, write $T_\ell ^{k} a$ for the matrix with components $(T^{k}_{\ell } a)_m^n = T^{jk}_{m\ell } a_j^n$. We then have the following expression:

$$\begin{aligned} f_{k}^\nu = {\sum }_{k_1 + k_2 = k} {\sum }_{\ell \le k_1}{\sum }_{\ell ' \le k_2} S_{\ell }^{k_1} \partial V_{\ell }\cdot S_{\ell '}^{ k_2} \partial V_{\ell '} - \widetilde{S}^{k_1,\nu }_{\ell } \partial \widetilde{V}^{\nu }_{\ell }\cdot \widetilde{S}^{k_2,\nu }_{\ell '} \partial V^{\nu }_{\ell '},\qquad \widetilde{V}^{\nu }_k = S_\varepsilon V^{\nu }_k. \end{aligned}$$

Using the commutator formulas (2.12), (4.14) twice, we have

$$\begin{aligned} g_k^\nu= & {} {\sum }_{\ell \le k-1} \delta ^{ij}\big (\partial _i S^{i'k}_{j\ell } \partial _{i'} h_\ell - \partial _i \widetilde{S}^{i'k,\nu }_{j\ell }\partial h_\ell ^\nu + S^{i'k}_{j\ell } \partial ^2_{ii'} h_\ell -\widetilde{S}^{i'k,\nu }_{j\ell } \partial ^2_{ii'}h_\ell ^{\nu }\big )\\&+ {\sum }_{\ell \le k-1} {\sum }_{\ell ' \le \ell -1} \delta ^{ij}\big ( S_{i\ell }^{i'k} \partial _{i'} S_{j\ell '}^{j'\ell } \partial _{j'} h_{\ell '} -\widetilde{S}_{i\ell }^{i'k,\nu } \partial _{i'} \widetilde{S}_{j\ell '}^{j'\ell ,\nu } \partial _{j'} h^\nu _{\ell '} \\&+ S_{i\ell }^{i'k} S^{j'\ell }_{j\ell } \partial _{i'j'}^2 h_{\ell '} - \widetilde{S}_{i\ell }^{i'k,\nu } \widetilde{S}^{j'\ell ,\nu }_{j\ell } \partial _{i'j'}^2h_{\ell '}^\nu \big ), \end{aligned}$$

where here we are writing $\partial = \partial _{x_0}$. Similarly, we have $F_k^{\nu } - F_k^{\nu -1} = f_k^{\nu ,\nu -1} + g_k^{\nu ,\nu -1}$, where

$$\begin{aligned}&f_{k}^{\nu ,\nu -1} = f_k^\nu - f_k^{\nu -1} = {\sum }_{k_1 + k_2 = k} {\sum }_{\ell \le k_1}{\sum }_{\ell ' \le k_2} \widetilde{S}_{\ell }^{k_1,\nu } \partial \widetilde{V}_{\ell }^\nu \cdot \widetilde{S}_{\ell '}^{ k_2,\nu } \partial V^\nu _{\ell '} \\&\quad -\widetilde{S}_{\ell }^{k_1,\nu -1} \partial \widetilde{V}_{\ell }^{\nu -1}\cdot \widetilde{S}_{\ell '}^{ k_2,\nu -1} \partial V^{\nu -1}_{\ell '}, \end{aligned}$$

and

$$\begin{aligned}&g_k^{\nu ,\nu -1} = g_k^{\nu } - g_k^{\nu -1} = {\sum }_{\ell \le k-1} \partial \widetilde{S}^{k,\nu }_{\ell } \partial h^\nu _\ell - \partial \widetilde{S}^{k,\nu -1}_\ell \partial h_\ell ^{\nu -1} \nonumber \\&\quad + \widetilde{S}^{k,\nu }_\ell \partial ^2 h^\nu _\ell -\widetilde{S}^{k,\nu -1}_\ell \partial ^2 h_\ell ^{\nu -1}\nonumber \\&\quad + {\sum }_{\ell \le k-1} {\sum }_{\ell ' \le \ell } \widetilde{S}^{\ell ,\nu }_{k-1} \partial \widetilde{S}^{\ell ',\nu }_\ell \partial h^\nu _{\ell '} - \widetilde{S}^{\ell ,\nu -1}_{k-1} \partial \widetilde{S}^{\ell ',\nu -1}_\ell \partial h^{\nu -1}_{\ell '} \nonumber \\&\quad + \widetilde{S}^{\ell ,\nu }_{k-1} \widetilde{S}^{\ell ',\nu }_{\ell } \partial ^2 h^\nu _{\ell '} - \widetilde{S}^{\ell ,\nu -1}_{k-1} \widetilde{S}^{\ell ',\nu -1}_{\ell } \partial ^2 h_{\ell '}^{\nu -1}. \end{aligned}$$

(E.23)

We first consider the case $r-k-2 \ge 2$. After performing the usual manipulations and using that $H^{r-k-2}$ is an algebra, to control $||f_k^\nu ||_{H^{r-k-2}}$, it suffices to prove that for $k' \le k, \ell \le k$, writing $j =r-k-2$

$$\begin{aligned} ||S^{k'}_\ell ||_{H^{j}} + ||\widetilde{S}^{k',\nu }_\ell ||_{H^{j}} + ||\partial V_\ell ||_{H^{j}} + {\sum }_{\alpha = 0,1} ||\partial S_\varepsilon ^\alpha V_\ell ||_{H^j} \le K(E_0, m_{k}^\nu ), \end{aligned}$$

(E.24)

and that, with $K' = K'(E_0, m_{k}^\nu , m_k^{\nu -1})$,

$$\begin{aligned}&||S^{k'}_\ell - \widetilde{S}^{k',\nu }_\ell ||_{H^{j}} + ||\partial V_\ell - \partial V_\ell ^\nu ||_{H^{j}} + ||\partial V_\ell - \partial \widetilde{V}^{\nu }_\ell ||_{H^{j}}\nonumber \\&\quad \le K' \big ( {\sum }_{\ell \le k} ||V_\ell - V_\ell ^\nu ||_{H^{j+1}} + ||u_{-1}^\nu ||_{H^{j+2}} + \varepsilon E_0\big ). \end{aligned}$$

(E.25)

The first two terms in (E.24) are bounded by the right side of (E.24) by Lemma E.7 and the other terms are bounded by the right side using the definition of the $m_k^\nu$ and the fact that $S_\varepsilon$ is bounded on Sobolev spaces.

The first term in (E.25) is bounded by the right-hand side of (E.25) by Lemma E.7, and the second term is directly bounded by $||V_\ell - V_\ell ^\nu ||_{H^{r-k-1}}$. To control the third term, we write $\widetilde{V}_\ell ^\nu = S_\varepsilon V_\ell ^\nu = S_\varepsilon V_\ell + S_\varepsilon (V_\ell ^\nu - V_\ell )$. Using that $||(1 - S_\varepsilon ) \partial V_\ell ||_{H^{r-k-2}} \le C\varepsilon ||V_\ell ||_{H^{r-k}}$ and $||S_\varepsilon (V_\ell - V_\ell ^\nu )||_{H^{s-k-1}} \le C||V_\ell - V_\ell ^\nu ||_{H^{r-k-1}}$ gives the bound for $f_k^\nu$. The bound for $f_k^{\nu ,\nu -1}$ follows in a nearly identical way. The case $r -k-2 \le 1$ is similar and follows the same lines as the proof of e.g. ().

We now bound $g_k^\nu$. We just prove estimates for the terms on the first line of (E.23) as the terms on the second line can be bounded in a similar manner. It suffices to prove that, for $\ell \le k-1$ with $j = r-k-2$,

$$\begin{aligned}&{\sum }_{m = 0,1} ||\partial ^m S^k_\ell ||_{H^{j}} + ||\partial ^m \widetilde{S}^{k,\nu }_\ell ||_{H^{j}} + ||\partial ^{m+1} h_\ell ||_{H^{j}} + ||\partial ^{m+1} h_\ell ^\nu ||_{H^{j}} \le K(E_0, m_k^\nu ), \\&{\sum }_{m = 0,1} ||\partial ^m {}_{\!}S^k_\ell \!- \partial ^m{}_{\!}\widetilde{S}^{k,\nu }_\ell \!||_{H^{j}}\! + ||\partial ^{m+1}\! h_\ell \! \\&\quad - \partial ^{m+1} \!h_\ell ^\nu ||_{H^{j}} \!\le \! K'\big ( {\sum }_{\ell ' \le k} ||V_{\!\ell '} \!- \!V_{\!\ell '}^{\!\nu }||_{H^{j+1}} + ||u_k^{\nu }||_{H^{j+2}} + ||u_{-1}^{\nu }\!||_{H^{j+2}}\!\big ). \end{aligned}$$

These estimates follow from Lemma E.7 since $h_k^\nu \!= h_k + u_k^\nu$. The estimates for $g_k^\nu \!- g_k^{\nu -1}\!\!$ are similar. $\quad \square$

To complete the proof of the estimates for $F_k-F_k^\nu$, we need the following two estimates to relate $V_k, V_k^\nu$ to the initial data $V_0, h_0$ and the perturbations $u^\nu$. We need a bit more notation. Given a diffeomorphism $X:\Omega \rightarrow X(\Omega )$ and a function $f:\Omega \rightarrow \mathbb {R}$, let $\Phi [X,f] = \phi \circ X^{-1}$, where $\phi$ is defined by

$$\begin{aligned} (X, f) \mapsto \phi (x) = \int _{X(\Omega )} |x-x'|^{-1} \rho (f(x'))\, \mathrm{d}x', \quad x \in \mathbb {R}^3. \end{aligned}$$

Set $\hat{x} = x_0 + t \sum _{k \ge 0}V_k t^k/(k+1)!$, $\hat{x}^\nu (t) = x_0 + t \sum _{k \ge 0} V_k^\nu t^k/(k+1)!$ and write $x_\ell = D_t^\ell \hat{x}|_{t = 0}, x_\ell ^\nu = D_t^\ell \hat{x}^\nu |_{t = 0}$. Set $\phi _\ell = D_t^\ell \Phi [\hat{x},\hat{h}]|_{t = 0}$ and $\phi _\ell ^\nu = D_t^\ell \Phi [\hat{x}^\nu , \hat{h}^\nu ]|_{t = 0}$. Then

Lemma E.9

With notation as in the previous lemma, for each $k = -1,..., r-2$, there are continuous functions $K_0 = K_0(E_0), K_0' = K_0'(E_0, ||u_{-1}^\nu ||_{H^{r-k+1}}, {{\,\mathrm{{\textstyle {\sum }}}\,}}_{\ell \le k-1} ||u^\nu _{\ell }||_{H^{r-k+1}})$, $K_0'' = K_0''(E_0, ||u^\nu ||_r, ||u^{\nu -1}||_r)$ so that

$$\begin{aligned}&||V_k||_{H^{r-k}} \le K_0, \quad ||V_k^\nu ||_{H^{r-k}} \le K_0', \quad ||V_k - V_k^\nu ||_{H^{r-k}} \le K_0' \big ( ||u^\nu _{-1}||_{H^{r-k+1}} \\&\quad + {\sum }_{\ell \le k-1} ||u^\nu _\ell ||_{H^{r-k+1}}\big ), \end{aligned}$$

and with $U^\nu = u^\nu - u^{\nu -1}$,

$$\begin{aligned} ||V_k^\nu - V_k^{\nu -1}||_{H^{r-k}} \le K_0'' \big ( ||U^\nu _{-1}||_{H^{r-k+1}} + {\sum }_{\ell \le k-1} ||U^\nu _\ell ||_{H^{r-k+1}}\big ). \end{aligned}$$

Proof

These estimates all follow from the definitions of $V_k, V_k^\nu$ in (E.19), (E.20), the estimates for $S, \widetilde{S}$ from Lemma E.7, and the estimates for $\phi _{k-1}, \phi _{k-1}^\nu$ in Lemma E.10.$\quad \square$

It still remains to control $\phi _{k-1}, \phi _{k-1}^\nu$. We shall not use this observation to prove estimates, but we remark that we have the following explicit representation formula for $\phi _{k-1}$:

$$\begin{aligned} \phi _{k-1}(y) = {\sum }_{k_1 + \dots k_j + k' \le k-1} \int _{\Omega } K_{k_1,\dots , k_j}(y,y') J_{k'}(y')\, \mathrm{d}y', \end{aligned}$$

where, for some constants $d_{k_1\cdots k_j}$,

$$\begin{aligned} K_{k_1,\dots , k_j}(y,y') =d_{k_{\!1}{\!}\dots {\!} k_{\!j}}\frac{\!\!(\delta x_{k_1\!}\cdot _{\!} \delta x_{k_2\!}) \cdots (\delta x_{k_{\!j\!-\!1}}\cdot \delta x_{k_j})\!\! }{|x_0(y)- x_0(y')|}, \qquad J_{k'} = D_t^{k'} (\rho (\hat{h}) \hat{\kappa })|_{t = 0}, \end{aligned}$$

with $\hat{\kappa } = \det (\partial y/ \partial \hat{x})$, and where we are writing $\delta W(y,y) = (W(y) -W(y'))/|x_0(y)- x_0(y')|$. Similarly,

$$\begin{aligned} \phi ^\nu _{k-1}(y) = {\sum }_{k_1+ \cdots k_j + k' \le k-1} \int _{\Omega } K_{k_1,..., k_j}^\nu (y,y') J_{k'}^\nu (y')\, \mathrm{d}y', \end{aligned}$$

where

$$\begin{aligned}&K_{k_1,\dots , k_j}^\nu (y,y') = d_{k_{\!1}{\!}\dots {\!} k_{\!j}} \frac{\!\!(\delta x^\nu _{k_1\!}\cdot _{\!} \delta x^\nu _{k_2\!}) \cdots (\delta x^\nu _{k_{\!j\!-\!1}}\cdot \delta x^\nu _{k_j})\!\! }{|x_0(y)- x_0(y')|},\\&\qquad J_{k'}^\nu = D_t^{k'} (\rho (\hat{h^\nu }) \hat{\kappa ^\nu })|_{t = 0}, \qquad \widehat{\kappa }^\nu = \det (\partial \hat{x}^\nu /\partial y). \end{aligned}$$

Lemma E.10

With notation as in Lemma E.8, for each $k = 0,..., r-2$, there are continuous functions $K_0 = K_0(E_0)$, $K_0' = K_0'(E_0, m_{k-1}^\nu )$, $K_0'' = K_0''(E_0, ||u^\nu ||_s, ||u^{\nu -1}||_s)$ so that

$$\begin{aligned}&||\phi _{k-1}||_{H^{r-k+1}} \le K_0, \qquad ||\phi _{k-1}^\nu ||_{H^{r-k+1}} \le K_0'. \\&||\phi _{k-1} - \phi _{k-1}^\nu ||_{H^{r-k+1}} \le K_0' {\sum }_{\ell \le k-1} ||u_{\ell }^\nu ||_{H^{r-k}},\\&\qquad ||\phi _{k-1}^\nu - \phi _{k-1}^{\nu -1}||_{H^{r-k+1}} \le K_0^{''} {\sum }_{\ell \le k-1} ||U_{\ell }^\nu ||_{H^{r-k}}. \end{aligned}$$

Proof

The estimates follows from Theorem 7.5, respectively Theorem 7.9. $\quad \square$

Appendix F: Existence and Estimates for the Wave Equations

1.1 F.1: Existence for the Linear Wave Equations

Fixing $r \ge 7, t_1> 0, V \in \mathcal {X}^{r+1}(t_1)$ and defining $\widetilde{\Delta }= \widetilde{\Delta }[V]$ as in (4.4), the goal of this section is to solve the linear wave equation

$$\begin{aligned} D_t^2 \varphi - \sigma \widetilde{\Delta }\varphi&= F, \quad \text { in } [0,t_1]\times \Omega ,\quad \text {with}\quad \varphi =0, \text { on } [0,t_1]\times \partial \Omega , \end{aligned}$$

(F.1)

$$\begin{aligned} \varphi (0,y)&= \varphi _0(y),\quad D_t\varphi (0,y) = \varphi _1(y), \quad \text { in } \Omega , \end{aligned}$$

(F.2)

where $\sigma = \sigma (t,y)$ satisfies $0< c_0 < \sigma \le c_1$ for some constants $c_0, c_1$. We will omit the dependence on $c_0, c_1$ in what follows. Compared with the estimates in Section 6, we have divided by $\sigma$ and abused notation slightly to make the following computations simpler.

As in Section 4.3, there are compatibility conditions for (F.1)–(F.2). We say $\varphi _0, \varphi _1$ satisfy the compatibility condition to order s if there is a formal power series in t, $\hat{\varphi } = \sum t^k \varphi _k$, satisfying (F.1)–(F.2) and

$$\begin{aligned} \varphi _k \in H^1_0(\Omega ), \qquad k = 0,...,s. \end{aligned}$$

(F.3)

Note that since $\hat{\varphi }$ satisfies (F.1), the coefficients $\varphi _k$ can be computed recursively from $\varphi _0, \varphi _1$ and time derviatives of F at $t = 0$:

$$\begin{aligned} \varphi _k = D_t^{k-2} \big ( \sigma \widetilde{\Delta }\widehat{\varphi } + F\big )\big |_{t = 0}. \end{aligned}$$

We will control solutions to (F.1)–(F.2) using the quantities

$$\begin{aligned} Y_s(t) = \Big ({\sum }_{k \le s} \int _\Omega |D_t^{k+1} \varphi |^2 + \sigma \delta ^{ij} (D_t^{k}\widetilde{\partial }_i \varphi )(D_t^{k}\widetilde{\partial }_j \varphi ) \, \mathrm{d}y\Big )^{1/2}. \end{aligned}$$

The main result we need is

Proposition F.1

Fix $r\! \ge \!7$ and $t_1\ge 0$, and suppose that $V\! \!\in \! \mathcal {X}^{r+{}_{\!}1}(t_1)\!$ satisfies (9.4). Suppose also that

$$\begin{aligned} \widetilde{x}&\in L^\infty ([0,t_1] ; H^r(\Omega )), \nonumber \\ D_t \widetilde{x}\in L^\infty ([0,t_1]; H^r(\Omega )),\quad \text {and}\quad D_t^k D_t \widetilde{x}&\in L^\infty ([0,t_1]; H^{r-k+1}(\Omega )),&k = 1,..., r+1,\nonumber \\ D_t^k\sigma&\in L^\infty ([0, t_1];H^{r-k}(\Omega )),&k = 0,..., r, \end{aligned}$$

(F.4)

and that the bound (5.2) holds. Also assume that for some s with $0 \le s \le r$,

$$\begin{aligned} F\in L^\infty ([0,t_1]; H^{s-1}(\Omega )),\quad \text {and}\quad D_t^k F&\in L^\infty ([0,t_1]; H^{s-k}(\Omega )),&k = 1,..., s, \end{aligned}$$

(F.5)

and that the compatibility condition (F.3) holds for $k = 0,...,s$. Take $K = K_{s,r}$ so that

$$\begin{aligned} {\sup }_{\,0 \le t \le t_1\,}\big ( ||\widetilde{x}(t)||_{r} + ||V(t)||_{r} + ||D_t V(t)||_r + ||\sigma (t)||_r + ||F(t)||_{s-1}\big ) \le K. \end{aligned}$$

(F.6)

Then the problem (F.1)–(F.2) has a unique solution $\varphi$ satisying

$$\begin{aligned} D_t^{s+1} \varphi \in L^\infty ([0,t_1]; L^2(\Omega )), \qquad D_t^{\ell } \widetilde{\partial }\varphi \in L^\infty ([0,t_1]; H^{s-\ell }(\Omega )),&\ell = 0,..., s, \end{aligned}$$

(F.7)

and there are continuous functions ${\mathcal {C}}_s$ depending on $M, Y_{s-1}(0), t_1,$ and K so that

$$\begin{aligned} {\sup }_{\,0 \le t \le t_1} Y_s(t) \le {\mathcal {C}}_s\Big (Y_s(0) + \int _0^{t_1} ||F(\tau )||_{s,0} \,d\tau \Big ), \end{aligned}$$

(F.8)

and for $0 \le t \le t_1$

$$\begin{aligned}&||\widetilde{\partial }\varphi (t)||_{s} \le {\mathcal {C}}_s ( Y_s(t) + ||F(t)||_{s-1} ), \qquad ||\widetilde{\partial }\varphi (t)||_{r} \le {\mathcal {C}}_r \big ( Y_r(t) \\&\quad + ||F(t)||_{r-1} + \varepsilon ^{-1}(||J_\varepsilon x(t)||_r + 1)Y_{r-1}(t) \big ). \end{aligned}$$

This result is well-known (see e.g. [10] or [9]) and will follow from a Galerkin method. However, we will need to be careful about the regularity of $\widetilde{x}$ and we will use our elliptic estimates from Section B in place of “standard” elliptic estimates. We do not claim that this result is optimal with respect to the total number of derivatives of $\widetilde{x}, V, D_tV,\sigma$ required and in many of the following results it is obvious that one can do with much weaker assumptions on these variables. We start by constructing weak solutions to the system (F.1)–(F.2). Let $\{e_k\}_{k = 0}^\infty$ be the $L^2$-normalized eigenfunctions in $H^1_0(\Omega )$ of the Dirichlet Laplacian in the y-coordinates $\Delta _y \!= \partial _{1}^2\! + \partial _{2}^2\! + \partial _{3}^2$. Let $d_{m}^k\! \in \! C^2([0,t_1])$, $k \!=\! 1,..., m$ solve the following system:

$$\begin{aligned} D_t^2 d_{m}^k + B_{k}(d_{m})&= \int _\Omega Fe_k\,\mathrm{d}y,&k =1,..., m, \nonumber \\ d_{m}^k(0) = (\varphi _0, e_k), \qquad D_t d_{m}^k(0)&= (\varphi _1, e_k),&k = 1,..., m, \end{aligned}$$

(F.9)

where

$$\begin{aligned} B_{k}(d_{m}) = {\sum }_{\ell = 1}^k d_{m}^{\,\ell } \int _\Omega \sigma \delta ^{ij} (\widetilde{\partial }_i e_\ell ) (\widetilde{\partial }_j e_k)\, \mathrm{d}y. \end{aligned}$$

Define

$$\begin{aligned} \varphi ^{m}(t) = {\sum }_{k = 1}^m d_{m}^k(t) e_k. \end{aligned}$$

(F.10)

Multipling (F.9) by $d_{m}^k$, summing over $k \le m$ and using (F.10), we have

$$\begin{aligned} \int _\Omega D_t^2 \varphi ^{m}e_k \, \mathrm{d}y + \int _\Omega \sigma \delta ^{ij} (\widetilde{\partial }_i \varphi ^{m})(\widetilde{\partial }_j e_k)\, \mathrm{d}y = \int _\Omega Fe_k\, \mathrm{d}y,&k=1,..., m. \end{aligned}$$

(F.11)

We now prove the basic energy estimate

Lemma F.2

If $\varphi ^{m}$ is as above, there is a constant $C_0 = C_0(M, K)$ so that

$$\begin{aligned}&{\max }_{\,0 \le t \le t_1} \big (||D_t \varphi ^{m}(t)||_{L^2(\Omega )} + ||\widetilde{\partial }\varphi ^{m}(t)||_{L^2(\Omega )}\big ) + || D_t^2 \varphi ^{m}||_{L^2(0,t_1; H^{-1}(\Omega ))}\nonumber \\&\quad \le C_0\big ( ||\varphi _0||_{H^1(\Omega )} + ||\varphi _1||_{L^2(\Omega )} + ||F||_{L^2(0,t_1;L^2(\Omega ))}\big ). \end{aligned}$$

(F.12)

Proof

We multiply (F.11) by $D_t d_m^k$ and sum over $k = 1,..., m$ to get

$$\begin{aligned} \int _\Omega (D_t^2 \varphi ^{m}) (D_t \varphi ^{m}) \, \mathrm{d}y + \int _\Omega \sigma \delta ^{ij} (\widetilde{\partial }_i \varphi ^{m}) (\widetilde{\partial }_j D_t \varphi ^{m})\, \mathrm{d}y = \int _\Omega FD_t \varphi ^{m}\mathrm{d}y. \end{aligned}$$

This first term is ${2}^{-1}{d} \,|| D_t \varphi ^{m}||_{L^2(\Omega )}/{dt}$. We use (D.2) and write the second term as

$$\begin{aligned}&\int _\Omega \sigma \delta ^{ij}(\widetilde{\partial }_i \varphi ^{m})D_t (\widetilde{\partial }_j\varphi ^{m}) \mathrm{d}y - \int _\Omega \sigma \delta ^{ij}(\widetilde{\partial }_jV^\ell ) (\widetilde{\partial }_i \varphi ^{m})(\widetilde{\partial }_\ell \varphi ^{m}) \mathrm{d}y\\&\quad = \frac{1}{2} \frac{d}{dt} ||\sqrt{\sigma }\widetilde{\partial }\varphi ^{m}||_{L^2(\Omega )}^2 - \int _\Omega \delta ^{ij} (\widetilde{\partial }_i \varphi ^{m}) \big (\sigma (\widetilde{\partial }_j S_\varepsilon V^\ell )(\widetilde{\partial }_\ell \varphi ) + D_t \sigma \widetilde{\partial }_j \varphi ^{m}\big )\, \mathrm{d}y, \end{aligned}$$

and we can bound this last term by $C(M)(1 + ||D_t \sigma ||_{L^\infty (\Omega )}) ||\widetilde{\partial }\varphi ^{m}||_{L^2}^2$.

Writing $Y_{(m)}(t) = ||D_t \varphi ^{m}||_{L^2} + ||\sqrt{\sigma } \widetilde{\partial }\varphi ^{m}||_{L^2}$, we have shown that

$$\begin{aligned} \frac{1}{2} \frac{d}{dt} Y_{(m)}^2 \le C(M) \Big ( (1 +||D_t \sigma ||_{L^\infty (\Omega )} ) Y_{(m)}^2 + ||F||_{L^2} Y_{(m)}\Big ), \end{aligned}$$

and so using that ${d}(Y_{(m)})^2 \!/{dt} = 2 Y_{(m)} {d} Y_{(m)}/{dt}$, dividing both sides by $Y_{(m)}$ and multiplying by the integrating factor $e^{-C(M)(1 +||D_t \sigma ||_{L^\infty (\Omega )}) t}$, we get

$$\begin{aligned}&{\sup }_{\,0 \le t \le t_1} \big (|| D_t \varphi ^{m}(t)||_{L^{{}_{\!}2}}{}_{\!} + ||\sqrt{\sigma }\widetilde{\partial }\varphi ^{m}(t)||_{L^{{}_{\!}2}} \!\big ) \!\le \! C \Big ( || D_t\varphi ^{m}(0)||_{L^{{}_{\!}2}}{}_{\!} \\&\quad + ||\!\sqrt{\!\sigma }\widetilde{\partial }\varphi ^{m}(0)||_{L^{{}_{\!}2}}{}_{\!} + \!\!\int _0^{t_1} \!\!\!\! \!||F(\tau )||_{L^{{}_{\!}2}} d\tau \Big ). \end{aligned}$$

where $C = C(M,\sup _{0 \le t \le t_1}||\widetilde{x}||_{r} + ||\sigma (t)||_{L^\infty }, t_1)$. Using the orthogonality of the $e_k$, we have

$$\begin{aligned} || D_t \varphi ^{m}(0)||_{L^2(\Omega )} + ||\sqrt{\sigma } \partial \varphi ^{m}(0)||_{L^2(\Omega )} \le ||\varphi _1||_{L^2(\Omega )} + ||\sqrt{\sigma } \varphi _0||_{H^1(\Omega )}, \end{aligned}$$

which proves the first part of (F.12). We now control $||D_t^2 \varphi ^{m}||_{H^{-1}(\Omega )}$.

Let $v \in H^1_0(\Omega )$ so that $||v||_{H^1(\Omega )} = 1$, and split $v = v^1 \!\!+ v^2$ with $v^1$ in the span of $e_1,..., e_m$. Then we have

$$\begin{aligned} \langle D_t^2 \varphi ^{m}, v\rangle = \langle D_t^2\varphi ^{m}, v^1\rangle = ( D_t^2\varphi ^{m}, v^1)_{L^2} = -(\sigma \widetilde{\partial }\varphi ^{m}, \widetilde{\partial }v^1)_{L^2} + (F, v^1)_{L^2}. \end{aligned}$$

The right-hand side is bounded by $C(M)( ||D_t\varphi ^{m}||_{L^2} + c_0||\widetilde{\partial }\varphi ^{m}||_{L^2(\Omega )} + ||F||_{L^2})||v^1||_{H^1(\Omega )}$. Noting that $||v^1||_{H^1(\Omega )} \le ||v||_{H^1(\Omega )} = 1$ and integrating in time gives the bound for $||D_t^2 \varphi ^{m}||_{L^2(0,t_1;H^{-1}(\Omega ))}$. $\quad \square$

Lemma F.3

With assumptions as in Proposition F.1, there is a unique $\varphi \! \in \! C([0,t_1];H^1_0(\Omega ))$ satisfying (F.1)–(F.2) with

$$\begin{aligned} D_t \varphi \in L^\infty (0,t_1;L^2(\Omega )), \qquad D_t^2 \varphi \in L^\infty (0,t_1; H^{-1}(\Omega )). \end{aligned}$$

Proof

By the uniform estimate (F.12) and Alaoglu’s theorem, passing to a subsequence we see that there is a $\varphi \in L^2(0,t_1; H^1_0(\Omega ))$ with $D_t \varphi \in L^2(0,t_1; L^2(\Omega )), D_t^2 \varphi \in L^2(0,t_1; H^{-1}(\Omega ))$ so that $\widetilde{\partial }\varphi ^{m}\rightarrow \widetilde{\partial }\varphi$ weakly in $L^2(0,t_1; L^2(\Omega ))$, $D_t \varphi ^{m}\rightarrow D_t \varphi$ weakly in $L^2(0,t_1; L^2(\Omega ))$ and $D_t^2 \varphi ^{m}\rightarrow \varphi$ weakly in $L^2(0,t_1; H^{-1}(\Omega ))$. Concretely, this means that if $v \in H^1_0(\Omega )$ then,

$$\begin{aligned}&\int _0^{t_1} \!\!\!\!\int _{\Omega }\! (\widetilde{\partial }^k\! \varphi ^{m}(t,y)) \widetilde{\partial }^k\!v(t,y)\, \mathrm{d}y\mathrm{d}t \rightarrow \!\! \int _0^{t_1}\!\!\!\! \int _\Omega \! (\widetilde{\partial }^k \!\varphi (t,y)) \widetilde{\partial }^k\! v(t,y) \, \mathrm{d}y\mathrm{d}t,\\&\qquad k\!=\!0,\!1, \quad \text {and}\quad \langle D_t^2 \varphi ^{m}\!\!, v \rangle \rightarrow \langle D_t^2 \varphi , v \rangle . \end{aligned}$$

Now, given $v \in C^1([0,t_1]; H^1_0(\Omega ))$ of the form

$$\begin{aligned} v(t) = {\sum }_{k = 1}^M v_k(t) e_k, \end{aligned}$$

we multiply the weak formulation (F.11) by $v_k(t)$, sum over k and integrate over $[0,t_1]$ to get that

$$\begin{aligned} \int _0^{t_1} \langle D_t^2\varphi ^{m}, v\rangle \, dt + \int _0^{t_1} \int _\Omega \sigma \delta ^{ij}(\widetilde{\partial }_i \varphi ^{m})(\widetilde{\partial }_j v) \, \mathrm{d}y dt = \int _0^{t_1} \int _\Omega Fv \,\mathrm{d}y dt. \end{aligned}$$

(F.13)

Taking $m \rightarrow \infty$ and using the above limits, we get that, for v of the above form,

$$\begin{aligned} \int _0^{t_1} \langle D_t^2 \varphi , v\rangle \,dt + \int _0^{t_1} \int _\Omega \sigma \delta ^{ij}(\widetilde{\partial }_i \varphi ) (\widetilde{\partial }_j v) \, \mathrm{d}y dt = \int _0^{t_1} \int _\Omega Fv \, \mathrm{d}y dt. \end{aligned}$$

(F.14)

Since such v are dense in $L^2(0,t_1; H^1_0(\Omega ))$ this holds for any v in this space. Hence, for almost every t,

$$\begin{aligned} \langle D_t^2 \varphi (t), v(t)\rangle + \int _\Omega \sigma \delta ^{ij}(\widetilde{\partial }_i \varphi (t))(\widetilde{\partial }_j v(t)) \,\mathrm{d}y = \int _\Omega F(t) v(t) \mathrm{d}y,\qquad v \in H^1_0(\Omega ). \end{aligned}$$

By an approximation argument and the fundamental theorem of calculus, using that $\varphi$ and its time derivatives are all in $L^2$ in time, we also get that $\varphi \in C([0,t_1];L^2(\Omega ))$ and $D_t\varphi \in C([0,t_1];H^{-1}(\Omega ))$ (see [9]). Hence (F.2) makes sense. We now have to check that $\varphi (0) = \varphi _0$ and $D_t \varphi (0) = \varphi _1$. Let $v \in C^2([0,t_1]; H^1_0(\Omega ))$ be such that $v(t_1) = D_t v(t_1) = 0$ and integrate by parts twice in time in (F.14) to get

$$\begin{aligned}&\int _0^{t_1}\!\!\! \int _\Omega \!\! D_t^2 v(t) \varphi (t) \,\mathrm{d}y\mathrm{d}t + \int _0^{t_1}\!\!\! \int _\Omega \!\! \sigma \delta ^{ij}(\widetilde{\partial }_i \varphi (t))( \widetilde{\partial }_j v(t))\,\mathrm{d}y \mathrm{d}t\\&\quad = \int _0^{t_1}\!\!\! \int _\Omega \!\! F(t) v(t) \, \mathrm{d}y \mathrm{d}t + \int _\Omega \!\! v(0) D_t\varphi (0) \! - \! D_t v(0) \varphi (0) \, \mathrm{d}y. \end{aligned}$$

On the other hand we can integrate by parts twice in (F.13) and take $m \rightarrow \infty$ to also get

$$\begin{aligned}&\int _0^{t_1} \!\!\! \int _\Omega \!\! D_t^2 v (t) \varphi (t)\, \mathrm{d}y \mathrm{d}t + \int _0^{t_1}\!\!\! \int _\Omega \!\! \sigma \delta ^{ij}(\widetilde{\partial }_i \varphi (t)) ( \widetilde{\partial }_j v(t)) \,\mathrm{d}y \mathrm{d}t \\&\quad = \int _0^{t_1}\!\!\! \int _\Omega \!\! F(t) v(t)\, \mathrm{d}y \mathrm{d}t + \int _\Omega \!\! v(0) \varphi _1 \!-\! D_t v(0) \varphi _0 \, \mathrm{d}y. \end{aligned}$$

Comparing these expressions using that v(0) and $D_t v(0)$ are arbitrary, we have $\varphi (0)\! = \!\varphi _0$ and $D_t\varphi \! = \!\varphi _1$. $\quad \square$

We now want to show that we get improved regularity of $\varphi$ when $\varphi _0, \varphi _1$ and $F$ are more regular. The first step is to show that the coefficients $d_m^\ell$ are more regular in this case and for this we take time derivatives of the equation (F.1). We apply $n \le r-1$ time derivatives to (F.1) and write $D_t \widetilde{\Delta }\varphi = \delta ^{ij} \widetilde{\partial }_j (D_t \widetilde{\partial }_i \varphi ) - \delta ^{ij} (\widetilde{\partial }_j V^\ell )\widetilde{\partial }_\ell \widetilde{\partial }_i \varphi$. We write the result as

$$\begin{aligned} D_t^{n+2} \varphi -\delta ^{ij}\widetilde{\partial }_j (\sigma D_t^n \widetilde{\partial }_i \varphi ) -\delta ^{ij}\widetilde{\partial }_{j} (D_t \sigma D_t^{n-1} \widetilde{\partial }_i\varphi ) + \delta ^{ij}\widetilde{\partial }_{j'}\big ( (\widetilde{\partial }_j S_\varepsilon V^{j'})D_t^{n-1} \widetilde{\partial }_i \varphi \big ) = F^n, \end{aligned}$$

(F.15)

where

$$\begin{aligned} F^n= & {} D_t^n F - {\sum }_{s = 2}^{n} (D_t^s \sigma )(D_t^{n-s} \widetilde{\Delta }\varphi )\nonumber \\&+ {\sum }_{s = 1}^n \delta ^{ij}(D_t^s A_{\,\,j}^b) (D_t^{n-s} \partial _b \widetilde{\partial }_i \varphi ) \nonumber \\&+ \delta ^{ij}(\widetilde{\partial }_j \sigma + \widetilde{\partial }_j D_t \sigma ) D_t^n \widetilde{\partial }_i\varphi - \delta ^{ij} (\widetilde{\partial }_{j'} \widetilde{\partial }_j S_\varepsilon V^{j'}) D_t^{n-1}\widetilde{\partial }_i \varphi . \end{aligned}$$

(F.16)

We write (F.15) like this because the third and fourth terms have as many space derivatives of $\varphi$ as the second term but fewer time derivatives, and so we will need to integrate by parts in space and time to handle them. The terms in $F^k$ will be lower-order and can be bounded in $L^2$ directly.

Multiplying this by arbitrary $v \in H^1_0(\Omega )$ and integrating by parts leads to the equation

$$\begin{aligned}&\int _\Omega (D_t^{n+2} \varphi ) v\, \mathrm{d}y + \int _\Omega \sigma \delta ^{ij} (D_t^n \widetilde{\partial }_i \varphi ) (\widetilde{\partial }_j v)\, \mathrm{d}y\\&\quad + \int _\Omega \delta ^{ij} (D_t^{n-1}\widetilde{\partial }_i \varphi ) \big ( D_t \sigma (\widetilde{\partial }_j v) \widetilde{\partial }_j S_\varepsilon V^{j'})(\widetilde{\partial }_{j'} v)\big )\, \mathrm{d}y = \int _\Omega F^n v\, \mathrm{d}y. \end{aligned}$$

With $d_m^\ell$ defined by (F.9), suppose that $d_m^\ell \in C^n([0,t_1])$ for some $n \ge 1$ and define

$$\begin{aligned} B_k^n= & {} B_k^n(d_m,..., D_t^n d_m) = {\sum }_{\ell \le k} \int _\Omega \sigma \delta ^{ij} D_t^n(d_m^\ell \widetilde{\partial }_i e_\ell )\widetilde{\partial }_j e_k\, \mathrm{d}y,\\ C_k^n= & {} C_k^n(d_m,..., D_t^{n-1} d_m) = {\sum }_{\ell \le k} \int _{\Omega } \delta ^{ij} D_t^{n-1}( d_m^\ell \widetilde{\partial }_i e_\ell ) \big ( D_t \sigma \widetilde{\partial }_j e_k \\&\quad + (\widetilde{\partial }_j S_\varepsilon V^{j'})\widetilde{\partial }_{j'} e_k\big )\, \mathrm{d}y. \end{aligned}$$

Also let $F^n(d_m)$ be $F^n$ with $\varphi$ replaced by $\varphi ^{m}= {\sum }_{k \le m} d_m^k e_k$. Let $\dot{d}_m^1,..., \dot{d}_m^k$ solve the ODE

$$\begin{aligned}&D_t \dot{d}_m^k + B_k^n + C_k^n = \int _{\Omega } F^n(d_m) e_k\, \mathrm{d}y,\nonumber \\&\quad \dot{d}_m^k(0) = (\varphi _{n}, e_k)_{L^2(\Omega )}, \quad k = 1,..., m, \end{aligned}$$

(F.17)

where $\varphi _{n}$ is defined by (F.3). By the existence and uniqueness theorem for ODE, it follows that $\dot{d}_{m}^{\,k}(t)\! =\! D_t^{n} d_{m}^{\,k}(t)$ for $0\! \le \! t\! \le \!t_1$ and this implies that $d_{m}^{\,k} \!\!\in \!C^{n+1}(0, \!t_1)$.

Before proving that the sequence $\varphi ^{m}$ converges in stronger topologies, we will need to ensure that $\varphi$ satisfies the equation (F.1) almost everywhere. We start with

Lemma F.4

Suppose that the hypotheses of Proposition F.1 hold. Let $\varphi$ be as in Lemma F.3. If $\varphi _0 \in H^2(\Omega ),\varphi _1 \in H^1_0(\Omega )$ and $D_tF\in L^2(0,t_1;L^2(\Omega ))$, then we have the improved regularity

$$\begin{aligned}&D_t \varphi \in C([0, t_1]; H^1_0(\Omega )),\qquad D_t^2 \varphi \in L^\infty ([0,t_1]; L^2(\Omega )),\nonumber \\&\quad D_t^3\varphi \in L^\infty ([0,t_1]; H^{-1}(\Omega )), \end{aligned}$$

(F.18)

Proof

Take $n = 1$, multiply (F.17) by $D_t^2 d_{m}^{\,k}$, use $\dot{d}_m^k= D_t d_m^{\,k}$ and write

$$\begin{aligned}&\widetilde{\partial }_j e_k D_t^2 d_{m}^{\,k} = \widetilde{\partial }_j D_t^2 \varphi ^{m}= D_t^2 \widetilde{\partial }_j \varphi ^{m}\\&\quad - (D_t^2 A_{\,\,j}^a) \partial _a \varphi ^{m}- 2 (D_t A_{\,\,j}^a)D_t \partial _a \varphi ^{m}\equiv D_t^2 \widetilde{\partial }\varphi ^{m}+ R^1_j, \end{aligned}$$

which gives

$$\begin{aligned}&B^1_k D_t^2 d_{m}^{\,k} = \int _\Omega \sigma \delta ^{ij} (D_t \widetilde{\partial }_i \varphi ^{m}) (\widetilde{\partial }_j e_k D_t^2 d_{m}^k)\, \mathrm{d}y \\&\quad = \frac{1}{2}\frac{d}{dt}\Big (\int _{\Omega } \sigma \delta ^{ij} (D_t \widetilde{\partial }_i \varphi ^{m})( D_t\widetilde{\partial }_j \varphi ^{m})\, \mathrm{d}y\Big )\\&\quad -\int _\Omega (D_t \sigma ) \delta ^{ij} (D_t \widetilde{\partial }_i \varphi ^{m}) (D_t \widetilde{\partial }_j \varphi ^{m})\, \mathrm{d}y \\&\quad -\int _\Omega \sigma \delta ^{ij} D_t \widetilde{\partial }_i\varphi ^{m}\big ( (D_t^2 A_{\,\,j}^a) \partial _a \varphi ^{m}- 2 (D_t A_{\,\,j}^a)D_t \partial _a\varphi ^{m}\big ) \, \mathrm{d}y, \end{aligned}$$

and similarly,

$$\begin{aligned} C^1_k D_t^2 d_{m}^{\,k}= & {} \frac{dC_1}{dt} - \int _\Omega (\widetilde{\partial }_i \varphi ^{m})\big ((D_t^2 \sigma ) (D_t \widetilde{\partial }_j \varphi ^{m}) \\&\quad - D_t \big ( \sigma (\widetilde{\partial }_j S_\varepsilon V^{j'})\big ) (D_t \widetilde{\partial }_{j'} \varphi ^{m})\big )\, \mathrm{d}y \\&\quad -\int _\Omega \sigma \delta ^{ij} (D_t\widetilde{\partial }_i \varphi ^{m})\big (R_j^1 - (\widetilde{\partial }_j S_\varepsilon V^{j'})R_{j'}^1 \big ) \, \mathrm{d}y, \end{aligned}$$

where

$$\begin{aligned} C_{1} = C_{1}[\varphi ^{m}]= \int _\Omega (D_t \sigma )\delta ^{ij} (\widetilde{\partial }_i \varphi ^{m}) (D_t \widetilde{\partial }_j\varphi ^{m}) -\delta ^{ij}\sigma (\widetilde{\partial }_j S_\varepsilon V^{j'}) (\widetilde{\partial }_i\varphi ^{m})( D_t \widetilde{\partial }_{j'}\varphi ^{m}) \mathrm{d}y. \end{aligned}$$

By Sobolev embedding and (D.1),

$$\begin{aligned} ||D_t^2 A_{\,\,j}^a||_{L^\infty (\Omega )} + ||D_t A_{\,\,j}^a||_{L^\infty (\Omega )} + ||D_t \widetilde{\partial }_jS_\varepsilon V^\ell ||_{L^\infty (\Omega )}&\le C(M) ||\widetilde{x}||_r,\\ ||D_t \sigma ||_{L^\infty (\Omega )} + ||\widetilde{\partial }D_t \sigma |||_{L^\infty (\Omega )} + ||D_t^2 \sigma ||_{L^\infty (\Omega )}&\le C ||\sigma ||_r. \end{aligned}$$

With $Y^1_{\!\!m}\! = \! ||D_t^2 \varphi ^{m}||_{L^2(\Omega )} \!+\! ||\sqrt{\sigma } D_t \widetilde{\partial }\varphi ^{m}||_{L^2(\Omega )}$, the above calculation shows that

$$\begin{aligned} \frac{d}{dt} \Big ((Y^1_{m})^2 - C_1[\varphi ^{m}]\Big ) \le C(M, ||\widetilde{x}||_r) (1 + ||\sigma ||_r)\Big ( Y^1_{m} + Y_{m} + ||F_{m}^1||_{L^2(\Omega )}\Big )Y^1_{m}. \end{aligned}$$

Multiplying both sides by the integrating factor $e^{-C(M, ||\widetilde{x}||_r)(1 + ||\sigma ||_r) t}$, integrating, and then using that $C[\varphi ^{m}] \le C(M) ||\widetilde{x}||_r (\delta (Y^1_{m})^2 + \delta ^{-1} Y_{m}^2)$ for any $\delta > 0$, this implies that

$$\begin{aligned}&Y^1_{m}(t)^2 \le C(M, ||\widetilde{x}||_r, ||\sigma ||_r) \Big ( Y^1_{m}(0)^2 + Y_{m}(t)^2 \\&\quad + \int _0^t Y_{m}^1(\tau )^2 + Y_{m}(\tau )^2 + ||D_t F_{m}^1(\tau )||_{L^2(\Omega )}^2\, d\tau \Big ), \end{aligned}$$

and so by Grönwall’s integral inequality, this implies

$$\begin{aligned}&{\sup }_{\,0 \le t \le t_1} Y^1_{m}(t) \le C \Big ( Y_{m}^1(0) +{\sup }_{\,0 \le t \le t_1} Y_{m}(t) \\&\quad + \int _0^{t_1} (1 + ||\sigma ||_r)Y_{m}(\tau ) + ||F_{m}^1(\tau )||_{L^2(\Omega )}\, d\tau \Big ). \end{aligned}$$

Arguing as in the previous lemma, this implies that the sequence $D_t \varphi ^{m}$ has limit $\dot{\varphi }$ with

$$\begin{aligned} D_t \dot{\varphi } \in L^\infty (0,t_1;L^2(\Omega )),\qquad D_t^2 \dot{\varphi } \in L^\infty (0, t_1; H^{-1}(\Omega )). \end{aligned}$$

Since also $\varphi ^{m}\rightarrow \dot{\varphi }$ in $L^2$ by the previous lemma, it follows that $\varphi = \dot{\varphi }$ and in particular we get the first two statements in (F.18). To get that $D_t^3 \varphi \in L^\infty ([0,t_1]; H^{-1}(\Omega ))$, we argue as in the previous lemma. Also, since the compatibility conditions (F.3) hold, we have that $Y^1_{m}(0) \rightarrow Y^1(0) = ||\varphi _2||_{L^2(\Omega )} + ||\sqrt{\sigma (0)} \widetilde{\partial }\varphi _1||_{L^2(\Omega )}$. $\quad \square$

We can now prove that $\varphi$ has enough regularity that the elliptic estimates from the Section B hold.

Lemma F.5

If $\varphi _0\! \in \!H^2(\Omega ), \varphi _1 \!\in \! H^1_0(\Omega )$ and (5.2), (F.6) hold, there is a constant $C_1\! \!=\! C_1( M, K, t_1)$ so that

$$\begin{aligned}&\text {ess sup}_{\,0 \le t \le t_1\,} \big ( ||\varphi (t)||_{H^2(\Omega )} + ||D_t \varphi (t)||_{H^1_0(\Omega )} \nonumber \\&\quad + || D_t^2 \varphi (t)||_{L^2(\Omega )}\big ) + || D_t^3 \varphi ||_{L^2(0,t_1;H^{-1}(\Omega ))}\nonumber \\&\quad \le C_1 \big ( ||F||_{H^1(0,t_1; L^2(\Omega )} + ||\varphi _0||_{H^2(\Omega )} + ||\varphi _1||_{H^1(\Omega )}\big ). \end{aligned}$$

(F.19)

Proof

By the previous lemma, we already have the second, third and fourth estimates in (F.19) and it just remains to bound the first term. The point is that we do not yet know that the wave equation (F.1) holds almost everywhere so we cannot use the elliptic estimate (5.8). As in [9], will instead prove an elliptic estimate for the approximate solution $\varphi ^{m}$. We let $\{\lambda _\ell \}_{\ell = 0}^\infty$ be the eigenvalues of $\Delta$ on $H^1_0(\Omega )$. Multiplying both sides of (F.9) by $\lambda _\ell d_{(m)}^{\,\ell }$ and summing from $\ell = 1$ to m, we get that

$$\begin{aligned} \int _\Omega \sigma \delta ^{ij} (\widetilde{\partial }_i \varphi ^{m}) (\widetilde{\partial }_j \Delta \varphi ^{m})\, \mathrm{d}y = \int _\Omega (F- D_t^2 \varphi ^{m}) \Delta \varphi ^{m}\, \mathrm{d}y. \end{aligned}$$

Since $\Delta \varphi ^{m}= 0$ on $\partial \Omega$, we integrate by parts in the left-hand side and use the estimate (B.8), which gives

$$\begin{aligned} ||\widetilde{\partial }\varphi ^{m}||_{H^1(\Omega )} \le C(M) \big ( ||D_t^2 \varphi ^{m}||_{L^2(\Omega )} + ||D_t \widetilde{\partial }\varphi ^{m}||_{L^2(\Omega )} + ||\widetilde{\partial }\varphi ^{m}||_{L^2(\Omega )} + ||\varphi ^{m}||_{L^2(\Omega )}\big ). \end{aligned}$$

Since $\varphi \in H^2(\Omega )$, we now have that $\varphi$ solves the equation (F.1)- (F.2) a.e. in $[0,t_1]\times \Omega$. $\quad \square$

Proof of Proposition F.1

We argue by induction. We have just shown that the theorem holds for $s = 0,1$. We suppose that the theorem holds for $s = 1,..., n-1 \le r-1$ and we now assume that the compatibility conditions (F.3) hold for $s = 0,..., n$. By the inductive assumption, there is a unique $\varphi$ satisfying the equation (F.1)–(F.2) in the weak sense so that

$$\begin{aligned} D_t^{n} \varphi \in L^\infty ([0,t_1] ; L^2(\Omega )),\quad D_t^{n - \ell } \widetilde{\partial }\varphi \in L^\infty ([0,t_1]; H^{\ell -1}(\Omega )), \quad \ell = 0,...,n. \end{aligned}$$

(F.20)

Moreover, with $\varphi ^{m}$ as defined above, we have that $D_t^n \varphi ^{m}(t) \rightarrow D_t^s \varphi (t)$ in $L^2(\Omega )$ and $D_t^{n-\ell }\widetilde{\partial }\varphi ^{m}(t) \rightarrow D_t^{n-\ell } \widetilde{\partial }\varphi (t)$ in $H^{\ell -1}(\Omega )$ for $\ell = 0,...,n$ and $0 \le t \le t_1$. We multiply (F.9) by $D_t^{n+1} d_{m}^\ell$ and write

$$\begin{aligned} (\widetilde{\partial }_j e_k )D_t^{n+1} d^k_{m} = D_t^{n+1} \widetilde{\partial }_j \varphi ^{m}- {\sum }_{s = 1}^{n+1} (D_t^s A_{\,\,j}^a)D_t^{n+1-s}\partial _a \varphi ^{m}\equiv D_t^{n+1} \widetilde{\partial }_j \varphi ^{m}- R^n_j, \end{aligned}$$

(F.21)

and this leads to

$$\begin{aligned} B_{k}^n D_t^{n+1}\! d_{m}^k\!= & {} \frac{1}{2} \frac{d}{dt} \int _\Omega \!\!\sigma \delta ^{ij} (D_t^n \widetilde{\partial }_i \varphi ^{m}) D_t^n \widetilde{\partial }_j \varphi ^{m}\,\mathrm{d}y \\&\quad - \int _\Omega \! (D_t \sigma )\delta ^{ij} (D_t^n \widetilde{\partial }_i \varphi ^{m}) D_t^n \widetilde{\partial }_j \varphi ^{m}\,\mathrm{d}y \\&\quad - \int _\Omega \!\!\sigma \delta ^{ij} (D_t^n \widetilde{\partial }_i\varphi ^{m})R_j^n\, \mathrm{d}y,\\ C_{k}^n D_t^{n+1} d_{m}^k\!= & {} \frac{dC_n}{dt} -\int _\Omega (D_t^{n-1} \widetilde{\partial }_i \varphi ^{m})\big ( (D_t^2 \sigma ) (D_t^{n}\widetilde{\partial }_j \varphi ^{m}) \\&\quad - D_t \big (\sigma (\widetilde{\partial }_j S_\varepsilon V^\ell )\big ) (D_t^{n} \widetilde{\partial }_\ell \varphi ^{m})\big )\, \mathrm{d}y\\&\quad -\int _\Omega \sigma \delta ^{ij} (D_t^n \widetilde{\partial }_i \varphi ^{m})\big (R_j^n - (\widetilde{\partial }_j S_\varepsilon V^{j'}) R_{j'}^n\big )\, \mathrm{d}y, \end{aligned}$$

where

$$\begin{aligned} C_{n} = \int _\Omega (D_t \sigma ) \delta ^{ij} (D_t^{n-1} \widetilde{\partial }_i \varphi ^{m}) (D_t^{n} \widetilde{\partial }_j \varphi ^{m}) - \delta ^{ij}\sigma (\widetilde{\partial }_j S_\varepsilon V^\ell )(D_t^{n-1} \widetilde{\partial }_i \varphi ^{m}) (D_t^{n}\widetilde{\partial }_\ell \varphi ^{m})\,\mathrm{d}y. \end{aligned}$$

Using Lemma F.6 to control $R^n, F^n$ and arguing as in the proof of Lemma F.4, we get

$$\begin{aligned} \frac{d}{dt} \Big (Y^n_{m}(t) -C_n[\varphi ^{m}]\Big ) \le C(M, ||\widetilde{x}||_r, ||D_t \widetilde{x}||_r, ||D_t^2 \widetilde{x}||_r, ||\sigma ||_r) \Big ( Y^n_{m} + Y_{m}^{n-1} + ||F^n_{m}||_{L^2(\Omega )}\Big ), \end{aligned}$$

and so applying the inductive assumption (F.20) and arguing as in the previous lemma, we get the result. $\quad \square$

Lemma F.6

Fix $r \ge 7$. Let $F^n_{m}$ be $F^n$ (defined in (F.16)) with $\varphi$ replaced by $\varphi ^{m}$ and $R^{n}$ be as in (F.21). There are constants $C_r$ depending on $M, ||\widetilde{x}||_r, ||D_t \widetilde{x}||_r, ||D_t^2 \widetilde{x}||_r, ||\sigma ||_r,$ so that if $k \le r$, then

$$\begin{aligned} ||F^n_m||_{L^2(\Omega )} + ||R_{}^{n}||_{L^2(\Omega )}&\le C_r\big (||\varphi ^{m}||_{n} + ||\widetilde{\partial }\varphi ^{m}||_{n} + ||D_t^n F||_{L^2(\Omega )}\big ). \end{aligned}$$

(F.22)

We remark that unlike the estimates in Section 6, these estimates depend on $||D_t^2 \widetilde{x}||_{r}$. This is because the estimates in that section are all in terms of $\widetilde{\partial }\varphi$ i.e. we estimate $||D_t^k \widetilde{\partial }\varphi ||_{L^2(\Omega )}$, but in the above proof we are forced to consider what amounts to $||\widetilde{\partial }D_t^k \varphi ^{m}||_{L^2(\Omega )}$. The error term this generates can be dealt with since in the application we have in mind, $D_t^2 \widetilde{x}= D_t S_\varepsilon V$ behaves like $\widetilde{\partial }\varphi$.

Proof

First, we control the first two terms in $F^n$ with $\varphi$ replaced by $\varphi ^{m}$. When $s \le r-2$, we have

$$\begin{aligned} ||D_t^s \sigma ||_{L^\infty (\Omega )} ||D_t^{n-s}\widetilde{\Delta }\varphi ^{m}||_{L^2(\Omega )} \le ||\sigma ||_r ||D_t^{n-s} \widetilde{\Delta }\varphi ^{m}||_{L^2(\Omega )}. \end{aligned}$$

To control this second term, we use the commutator estimate (D.9) to get

$$\begin{aligned} ||D_t^{n-s} \widetilde{\Delta }\varphi ^{m}||_{L^2(\Omega )} \le C(M, ||\widetilde{x}||_r) ||D_t^{n-s} \widetilde{\partial }\varphi ^{m}||_{H^1(\Omega )}. \end{aligned}$$

Since $s \ge 2$, we have $||D_t^{n-s} \widetilde{\partial }\varphi ^{m}||_{H^1(\Omega )} \le ||\widetilde{\partial }\varphi ^{m}||_{n-1}$. If instead $s = r-1, r$, the result is bounded by

$$\begin{aligned} ||D_t^s\sigma ||_{L^2(\Omega )} ||D_t^{n-s} \widetilde{\Delta }\varphi ^{m}||_{L^\infty (\Omega )} \le C ||\sigma ||_r ||D_t^{n-s} \widetilde{\Delta }\varphi ^{m}||_{H^2(\Omega )}, \end{aligned}$$

and so again applying the commutator estimate, this term is bounded by the right-hand side of (F.22) provided $||\widetilde{\partial }\varphi ^{m}||_{n - s + 3} \le ||\widetilde{\partial }\varphi ^{m}||_{n}$, and this follows since $r \ge n \ge s$, $s = r-1, r$ and $r \ge 7$.

We now control the remaining terms from the definition of $F^n\!\!$. The last two terms are clearly bounded by the right-hand side of (F.22) so we just bound the terms in the sum. When $s \!\le \! r\!-\!3$, we bound the terms by

$$\begin{aligned}&||D_t^s A_{\,\,j}^b||_{L^\infty } ||D_t^{n-s} \partial _b \widetilde{\partial }_i\varphi ^{m}||_{L^2(\Omega )} \\&\quad \le ||A_{\,\,j}^b||_{s+2} ||D_t^{n-s} \widetilde{\partial }\varphi ^{m}||_{H^1(\Omega )} \le C(M)||\widetilde{x}||_r ||\widetilde{\partial }\varphi ^{m}||_{n-s+1}, \end{aligned}$$

and since $s \ge 2$, we have $n-s+1 \le n-1$ as required.

We now consider the remaining cases $r\!-\!2\!\le \!s\!\le \!r$. In these cases we instead bound the summands by

$$\begin{aligned} ||D_t^s A_{\,\,j}^b||_{L^2(\Omega )} ||D_t^{n-s} \partial _b \widetilde{\partial }_i \varphi ^{m}||_{L^\infty (\Omega )} \le C(M, ||\widetilde{x}||_r)||D_t \widetilde{x}||_r ||D_t^{n-s} \widetilde{\partial }\varphi ^{m}||_{H^3(\Omega )}, \end{aligned}$$

and since in this case $n -s + 3 \le n-1$ (because $r-2 \le s \le n$ and $r \ge 7$), this second factor is bounded by the right-hand side of (F.22) as well, and this completes the proof of the bounds for $F^n_{m}$.

We now control $R^{n}_{}$. This follows in the same way as the bounds we have just proved but note that we also need to consider the case $s = r+1$. This is the reason that $||D_t^2 \widetilde{x}||_r$ enters into the estimates. When $s \le r-3$ we argue as above and the result is that

$$\begin{aligned}&||D_t^s A_{\,\,j}^b||_{L^\infty (\Omega )} ||D_t^{n+1-s} \partial \varphi ^{m}||_{L^2(\Omega )} \\&\quad \le C ||A_{\,\,j}^b||_{s+2} ||\partial _b \varphi ^{m}||_{n+1-s} \le C(M, ||\widetilde{x}||_r) ||\partial _b \varphi ^{m}||_{n+1-s}. \end{aligned}$$

The remaining cases are $s = r-2,r-1,r,r+1$ and for these we bound the result by

$$\begin{aligned} ||D_t^s A_{\,\,j}^b||_{L^2(\Omega )} ||D_t^{n+1-s} \partial _b \varphi ^{m}||_{L^\infty (\Omega )} \le C(M) ||D_t^2 \widetilde{x}||_r ||\partial _y \varphi ^{m}||_{n}, \end{aligned}$$

where in the last step we used that $n+1-s \le n$ when $s \ge r-2$ for $r \ge 7$. We now need to re-write $\partial _b \varphi ^{m}= A_{\,\,b}^j \widetilde{\partial }_j \varphi ^{m}$ and we note that by similar arguments to the above, we have

$$\begin{aligned} ||A_{\,\,b}^j \widetilde{\partial }_j \varphi ^{m}||_n \le C(M, ||\widetilde{x}||_r) ||D_t \widetilde{x}||_r ||\widetilde{\partial }\varphi ^{m}||_n. \end{aligned}$$

$\square$

1.2 F.2: Existence for a Nonlinear Wave Equation

We assume that (2.11) hold and that $e\!:\! (0,\infty )\! \rightarrow \! \mathbb {R}$ is a function satisfying (2.11). In this section we prove that the nonlinear wave equation

$$\begin{aligned} e'(\varphi ) D_t^2 \varphi - \widetilde{\Delta }\varphi&= F\text { in } [0,t_1]\times \Omega ,\qquad \text {with}\qquad \varphi = 0 \text { on } [0,t_1] \times \partial \Omega , \end{aligned}$$

(F.23)

$$\begin{aligned} \varphi (0,y)&= \varphi _0(y) ,\qquad D_t \varphi (0,y) = \varphi _1(y)\text { on } \Omega \end{aligned}$$

(F.24)

has a unique strong solution $\varphi$ satisfying (F.7). We will construct a solution so that for some $L=L[\varphi ]<\infty$,

$$\begin{aligned} {{{\,\mathrm{{\textstyle {\sum }}}\,}}}_{k + |J| \le 3} |D_t^k \partial _y^J \widetilde{\partial }\varphi | +|D_t^k \varphi | \le L, \quad \text { in } [0,t_1]\times \Omega . \end{aligned}$$

(F.25)

We assume that $F\!= \!F_1\! + F_2$ where $F_1\! = F_1(t,y)$ is a function and $F_2 \!= F_2[\varphi ,D_t\varphi ](t,y)$ is a functional so that there are continuous functions $N_{\!s}\!=\!N_{\!s}(L[\varphi ],\!||\varphi ||_{s-\!1},||\varphi ||_{s,0})$, $N_{\!s}'\!=\!N_{\!s}'(L[\varphi ],\!||\varphi ||_{s-\!1}\!)$ so that

$$\begin{aligned}&||D_t^s\! F_2[\varphi ,D_t\varphi ]||_{L^2(\Omega )}\! \le N_{s}(||D_t^{s+1} \! \varphi ||_{L^2(\Omega )}\! + ||\varphi ||_{L^2(\Omega )}),\nonumber \\&\qquad ||F_2[\varphi ,D_t\varphi ]||_{s\!-\!1}\! \le N_s'||\varphi ||_{s}. \end{aligned}$$

(F.26)

We will additionally assume that if $\varphi ,\psi$ satisfy (F.25) there are continuous functions $\overline{N}_s, \overline{N}_s^{\prime }$ depending on $L[\varphi ],L[\psi ], ||\varphi ||_{s}, ||\psi ||_{s}$ with $\overline{N}_s$ depending also on $||\varphi ||_{s+1,0}, ||\psi ||_{s+1,0}$ so that with $\dot{\varphi }\!=\!D_t \varphi$, $\dot{\psi }\!=\!D_t \psi$,

$$\begin{aligned}&||D_t^s F_2[\varphi ,\dot{\varphi }] - D_t^s F_2[\psi ,\dot{\psi }]||_{L^2(\Omega )} \le \overline{N}_s||\varphi -\psi ||_{s+1,0},\nonumber \\&\qquad \quad ||F_2[\varphi ,\dot{\varphi }] - F_2[\psi ,\dot{\psi }]||_{s-1} \le \overline{N}_s^{\prime } ||\varphi -\psi ||_{s}. \end{aligned}$$

(F.27)

In Section 8 we take $F_2\! = e''(\varphi ) (D_t\varphi )^2\! + \rho [\varphi ]$, which satisfies these estimates. The energies we use are

$$\begin{aligned} Y_s(t) = \Big (\frac{1}{2}{\sum }_{ k \le s} \int _\Omega e'(\varphi ) |D_t^{k+1 }\varphi |^2 + \delta ^{ij}(D_t^k \widetilde{\partial }_i \varphi )(D_t^k\widetilde{\partial }_j \varphi ) \, \widetilde{\kappa }\mathrm{d}y\Big )^{1/2}. \end{aligned}$$

The initial data $\varphi _0, \varphi _1$ satisfy the compatibility conditions to order s for the problem (F.23)–(F.24) if there is a formal power series solution $\widehat{\varphi } = \sum t^k \varphi _k$ to (F.23) which additionally satisfies

$$\begin{aligned} \varphi _k \in H^1_0(\Omega ), \qquad k = 0,..., s. \end{aligned}$$

(F.28)

Theorem F.7

Fix $r \!\ge \! 7$ and suppose that $V\! \!\in \! \mathcal {X}^{r+{}_{\!}1}(T_1)$ for some $T_1 \!>\!0$ satisfies (9.4) and that the bound (5.2) holds. Take K so that

$$\begin{aligned} {\sup }_{\,0 \le t \le T_1\,}\big ( ||\widetilde{x}(t)||_{r} + ||V(t)||_{r} + ||D_t V(t)||_{r} + ||D_t F_1(t)||_{r-1} + ||F_1(t)||_{r-1}\big ) \le K. \end{aligned}$$

Suppose that (F.4)–(F.5), (2.11) and the compatibility conditions (F.28) hold for some $s \le r$. Let $L_0$ satisfy

$$\begin{aligned} {{{\,\mathrm{{\textstyle {\sum }}}\,}}}_{k + |J| \le 3} ||\partial _y^L \widetilde{\partial }\varphi _k||_{L^\infty (\Omega )} + ||\varphi _k||_{L^\infty (\Omega )} \le L_0. \end{aligned}$$

There is a continuous function $G_r'$ so that if $t_1$ satisfies

$$\begin{aligned} t_1G_r'(M, L_0, L_0^{-1}, Y_r(0), K, T_1) \le 1,\quad \text {and} \quad t_1\le T_1, \end{aligned}$$

the problem (F.23)–(F.24) has a unique solution $\varphi$ satisfying

$$\begin{aligned} D_t^s \varphi \in L^\infty ([0,t_1]; L^2(\Omega )),\quad D_t^{s+1-\ell } \widetilde{\partial }\varphi \in L^\infty ([0,t_1]; H^{\ell -1}(\Omega )), \quad \ell = 0,..., s+1, \end{aligned}$$

and there are constants ${\mathcal {C}}_s$ depending on $M, L_0,\,Y_{s}(0), K$, and $t_1$ so that the following estimates hold:

$$\begin{aligned}&Y_s(t) \le {\mathcal {C}}_s \Big ( Y_s(0) + \int _0^t ||F_1(\tau )||_{s,0} + ||F_1(\tau )||_{s-1} \, d\tau \Big ),\\&\qquad ||\widetilde{\partial }\varphi (t)||_{s} \le {\mathcal {C}}_s \big ( Y_s(t) + ||F_1(t)||_{s-1}\big ), \end{aligned}$$

for $0 \le t \le t_1$ and

$$\begin{aligned} {{{\,\mathrm{{\textstyle {\sum }}}\,}}}_{k + \ell \le 2} |\partial ^\ell D_t^k \varphi (t,y)| \le 2 L_0, \qquad \text { in } [0,t_1] \times \Omega . \end{aligned}$$

(F.29)

We will construct a solution to (F.23)–(F.24) by considering the sequence $\varphi ^\nu$, $\nu = 0,1,...$, defined by

$$\begin{aligned} \varphi ^0&= {\sum }_{k = 0}^s \varphi _k {t^k}\!/{k!} ,\nonumber \\ D_t^2 \varphi ^\nu -{e'(\varphi ^{\nu -1})^{-1}} \widetilde{\Delta }\varphi ^\nu&= {e'(\varphi ^{\nu -1})^{-1}} F^{\nu -1}, \text { in } [0,t_1] \times \Omega ,\nonumber \\&\quad \text {with}\quad \varphi ^\nu = 0\text { on } [0,t_1] \times \partial \Omega , \end{aligned}$$

(F.30)

$$\begin{aligned} \varphi ^\nu (0,y)&= \varphi _0(0, y),\qquad D_t\varphi ^\nu (0,y) = \varphi _1(0,y), \text { on } \Omega , \end{aligned}$$

(F.31)

with $F^{\nu -1} = F_1 + F_2[\varphi ^{\nu -1}]$ and where $\widehat{\varphi } = \sum t^k \varphi _k$ is a given formal power series solution to (F.23). Note that with this choice of $\varphi ^0$, we have that $D_t^j \varphi ^0|_{t = 0} = \varphi _j^\varepsilon$, $j \le s$. This system also has compatibility conditions which must be satisfied to construct a sufficiently regular solution. Given $\varphi ^{\nu -1}$, let $\widehat{\varphi ^\nu } = \sum t^k \varphi ^\nu _k$ be a formal power series solution to (F.30). Taking time derivatives of (F.30) and restricting to $t = 0$, we see that the coefficients $\varphi ^\nu _k$ must satisfy

$$\begin{aligned} \varphi ^\nu _k = \big ( {e'(\varphi ^{\nu -1})^{-1}} \big (D_t^{k-2} \widetilde{\Delta }\widehat{\varphi ^\nu } + D_t^{k-2}F^{\nu -1} + \mathcal {G}_k[\widehat{\varphi ^\nu }, \varphi ^{\nu -1}]\big )\big |_{t = 0}, \end{aligned}$$

(F.32)

where we are writing

$$\begin{aligned} \mathcal {G}_k[\widehat{\varphi ^\nu }, \varphi ^{\nu -1}] = D_t^{k-2} \big ( e'(\varphi ^{\nu -1}) D_t^2 \widehat{\varphi ^\nu } \big ) - e'(\varphi ^{\nu -1}) D_t^{k}\widehat{\varphi ^\nu }. \end{aligned}$$

The compatibility conditions for the system (F.30)–(F.31) are then the requirement that

$$\begin{aligned} \varphi ^\nu _k \in H^1_0(\Omega ), k = 0,...,s. \end{aligned}$$

Since $\varphi ^\nu _0 = \varphi _0, \varphi ^\nu _1 = \varphi _1$ and both of these sequences are defined recursively, from (F.3) and (F.32) it follows that $\varphi ^\nu _k = \varphi _k$ for all $\nu \ge 0$ and so the compatibility conditions for the approximate problem (F.30)–(F.31) are satisfied so long as the compatibility conditions (F.28) for the nonlinear problem (F.23)–(F.24) hold.

We now argue by induction to show that the above problem has a unique solution with bounds that hold uniformly in $\nu$. Let $X^r_{t_1}$ be closure of $C^\infty ([0,t_1]; C^\infty (\Omega ))$ with respect to the norm

$$\begin{aligned} ||\varphi ||_{X^r_{t_1}} = {\sup }_{\,0 \le t \le t_1} {\sum }_{s =0}^{r} ||D_t^{s+1} \varphi (t)||_{L^2(\Omega )} + ||D_t^{s}\widetilde{\partial }\varphi (t)||_{H^{r-s}(\Omega )}. \end{aligned}$$

Assume that for $\nu \ge 1$ we have a solution $\varphi ^{\nu -1} \in X^r_{t_1}$ which moreover satisfies (F.29). Writing

$$\begin{aligned} Y^{\nu -1}_s(t) = {\sum }_{k \le s} \Big (\frac{1}{2} \int _\Omega e'(\varphi ^{\nu -2}) |D_t^{k+1} \varphi ^{\nu -1}(t)|^2 + |D_t^s \widetilde{\partial }\varphi ^{\nu -1}|^2\, \widetilde{\kappa }\mathrm{d}y\Big )^{1/2}, \end{aligned}$$

by Proposition F.1, we have the estimate

$$\begin{aligned}&Y^{\nu -1}_s(t) \le {\mathcal {C}}_s \Big (Y_s(0) + \int _0^t ||D_t^{s-1} F_1(\tau )||_{L^2(\Omega )}\, d\tau \Big ),\nonumber \\&\qquad ||\widetilde{\partial }\varphi ^{\nu -1} ||_s \le {\mathcal {C}}_s \big ( Y_s + ||F_1||_{s-1}\big ), \quad s = 0 ,..., r, \end{aligned}$$

(F.33)

where here ${\mathcal {C}}_s$ depends on $M, Y_{s-1}(0)$, K and $\sup _{0 \le t \le t_1} ||e'(\varphi ^{\nu -1}(t))||_{r}$. Note that we are using that $Y_s^{\nu -1}(0) = Y_s(0)$ in (F.33). By these estimates, (F.26) and Lemma D.9 to control $e'(\varphi ^{\nu -1})$, we have

$$\begin{aligned} ||F^{\nu -1}||_{s, 0} + ||F^{\nu -1}||_{s-1} + ||\sigma (\varphi ^{\nu -1})||_{s,0} + ||\sigma (\varphi ^{\nu -1})||_{r} \le C_s(M, L_0, Y_r(0), K). \end{aligned}$$

By (F.1), there is a unique $\varphi ^\nu \! \in X^r_{t_1}$ satisfying (F.30)–(F.31) and so that (F.7) holds. By the above estimates and the inductive assumption we also have:

$$\begin{aligned} Y_s^\nu (t) \le {\mathcal {C}}_s \Big ( Y_s(0) + \int _0^{t_1} ||F_1(\tau )||_{s,0} + ||F_1(\tau )||_{s-1} \, d\tau \Big ), \end{aligned}$$

(F.34)

where ${\mathcal {C}}_{{}_{\!}s} \!=_{\!}{\mathcal {C}}_{{}_{\!}s}(\!M_{\!},_{\!} L_0, \!Y_{\!r}(_{\!}0_{\!}),_{\!} K_{\!})$ and we again are using that $Y_{\!s}(0)$ is independent of $\nu _{\!}$. We note that by Sobolev embedding, the estimate (F.34) and the estimate (6.7), just as in the proof of Corollary 6.4, we have that

$$\begin{aligned}&L^{\!\nu }\!(t) \equiv \!{{{\,\mathrm{{\textstyle {\sum }}}\,}}}_{k +_{\!} |_{\!}J| \le 3} |\partial _y^J\! D_t^k \widetilde{\partial }\varphi ^\nu \!(t,_{\!}y)| + |D_{\!t\,}^k \!\varphi ^\nu \!(t,_{\!}y)| \\&\quad \le L_0 + t_1P_0^\nu ,\qquad \text {where}\quad P_0^\nu \!\equiv P_0^{\nu \!}(M_{\!}, {\sup }_{0 \le t \le t_1} L^{\!\nu \!}(t),\! Y_5^{\!\nu \,}\!(_{\!}0_{\!}),_{\!}K), \end{aligned}$$

and so a continuity argument (see the proof of Corollary 6.4) gives that $\sup _{0 \le t \le t_1} L^\nu (t) \le 2L_0$ provided that $t_1(2L_0)^{-1}P_0^\nu (M, 2L_0, Y_5^\nu (0), K) \le 1$. Note that in fact $P_0$ is independent of $\nu$ since $Y_5^\nu (0)$ is.

The sequence $\varphi ^{\nu }$ is therefore uniformly bounded in $X_{T_0}^r$ for a fixed $T_0\! > \! 0$, and therefore there is a $\varphi \!\in \! X_{T_0}^r$ so that $\varphi ^\nu \! \!\! \rightarrow \! \varphi$ weakly. We now show that there is $T^*\!\! =\! T^*(M, L_0, Y_{\!r}(0), K) \!\le T_0$ so that if $T_1\!\! \le \! T^*\!$, then

$$\begin{aligned} ||\varphi ^{\nu } - \varphi ^{\nu -1}||_{X^0_{T_1}} \le 2^{-1} ||\varphi ^{\nu -1} - \varphi ^{\nu -2}||_{X^{0}_{T_1}}. \end{aligned}$$

(F.35)

Assuming that this holds for the moment, it follows that the sequence $\varphi ^\nu$ is a Cauchy sequence in $X_{T_1}^0$ and so converges strongly to some $\tilde{\varphi } \in X_{T_1}^{0}$. This limit has to coincide with the $\varphi$ above and in particular this shows that the $\varphi ^\nu$ converges strongly to $\varphi$, and so $\varphi$ satisfies the nonlinear equation (F.23).

To prove (F.35), we take $T^* \!\le \! T_0$ and set $\psi = \varphi ^\nu \!- \varphi ^{\nu -1}\!$ and note that with $F^{\nu ,\nu -1}\! = F_2^\nu \! - F_2^{\nu -1}\!$ we have

$$\begin{aligned}&e'(\varphi ^\nu ) D_t^2 \psi - \widetilde{\Delta }\psi = F^{\nu ,\nu -1} + (e'(\varphi ^\nu ) - e'(\varphi ^{\nu -1})) D_t^2 \varphi ^{\nu -1},\\&\quad \text {with}\quad \psi |_{[0,t_1]\times \partial \Omega } = 0, \, \quad \psi |_{t = 0} = D_t\psi |_{t = 0} = 0. \end{aligned}$$

By the estimates (F.27), the estimate (F.8), and the product estimate (A.25), we have that

$$\begin{aligned}&Y_0^{\nu ,\nu -1}(t)\equiv \Big (\frac{1}{2}\int _\Omega e'(\varphi ^\nu ) |D_t \psi |^2 + | \widetilde{\partial }\psi |^2\, \widetilde{\kappa }\mathrm{d}y \Big )^{1/2} \le {\mathcal {C}}_0 \int _0^{t_1} ||\varphi ^{\nu -1} \\&\quad - \varphi ^{\nu -2}||_{1}\, dt \le {\mathcal {C}}_0 t_1||\varphi ^{\nu -1} -\varphi ^{\nu -2}||_{X_{t_1}^0}, \end{aligned}$$

where ${\mathcal {C}}_{{}_{\!}s} \!=_{\!}{\mathcal {C}}_{{}_{\!}s}(\!M_{\!},_{\!} L_0, \!Y_{\!r}(_{\!}0_{\!}),_{\!} K_{\!})$. Since $||\varphi ^{\nu \!}\!\!- \!\varphi ^{\nu \!-\!1\!}||_{X^0_{t_1}} \! \!\lesssim _{\!}\sup _{\,0 \le t \le t_1\!} Y_0^{\nu _{\!},\nu -\!1\!}\!\!$, taking $t_1$ sufficiently small gives (F.35).

1.3 F.3: Proof of Estimates for the Wave Equation

Lemma F.8

For each $s \ge 0$, there is a continuous function $G'_s(t) \!=\! G'_s( M, ||\widetilde{x}(t)||_s, ||V(t)||_{\mathcal {X}^{s}},W_{s-1}(t))$ and a polynomial P so that if (6.1)–(6.6) hold, then

$$\begin{aligned} \frac{d}{dt} W_s\le G'_s \Big ( W_s + ||F_1||_{s,0} + ||F_1||_{s-1} + ||V||_{\mathcal {X}^{s+1}} + P(L, W_{s-1}, ||F||_{s-2})W_{s}\Big ). \end{aligned}$$

(F.36)

Proof

We start by showing that

$$\begin{aligned} \frac{d}{dt} W_s^2 \le G_s'' \Big ( W_s + ||F||_{s,0} + ||\widetilde{\partial }\varphi ||_{s} + ||V||_{\mathcal {X}^{s+1}} + P(L,||\varphi ||_s) W_s\Big ) W_s, \end{aligned}$$

(F.37)

for a continuous function $G_s'' = G''_s(M, ||\widetilde{x}||_s, ||V||_{\mathcal {X}^{s}})$. We have

$$\begin{aligned}&\frac{d}{dt} W_{\!\!s}^2 \!=\!\! ={\sum }_{k \le s} \int _\Omega \sigma (D_t^{k+2}\varphi )(D_t^{k+1} \varphi ) \\&\quad + \delta ^{ij} (D_t^k \widetilde{\partial }_i \varphi )\widetilde{\partial }_j ( D_t^{k+1}\varphi ) \, \widetilde{\kappa }\mathrm{d}y\\&\quad +\!{\sum }_{k \le s}\!\Big ( \int _\Omega \delta ^{ij} (D_t^k \widetilde{\partial }_i \varphi ) [\widetilde{\partial }_j, D_t^{k+1}] \varphi \, \widetilde{\kappa }\mathrm{d}y \\&\quad + \frac{1}{2}\int _\Omega (D_t \sigma ) (D_t^{k+1} \varphi )^2 + (D_t\log \widetilde{\kappa })\big ( (D_t^{k+1}\varphi )^2 +|D_t^k \widetilde{\partial }\varphi |^2)\, \widetilde{\kappa }\mathrm{d}y \Big ). \end{aligned}$$

The last line is bounded by $C(M)(1 + L) (W_s)^2$. Integrating by parts, the terms on the first line are

$$\begin{aligned}&\int _\Omega \big (\sigma D_t^{k+2} \varphi - \delta ^{ij} (\widetilde{\partial }_j D_t^k \widetilde{\partial }_i \varphi )\big )(D_t^{k+1} \varphi ) \\&\quad = \int _\Omega \big ( \sigma D_t^{k+2} \varphi - D_t^k \widetilde{\Delta }\varphi \big )(D_t^{k+1} \varphi ) + \int _\Omega \delta ^{ij}([D_t^k, \widetilde{\partial }_j]\widetilde{\partial }_i \varphi )( D_t^{k+1} \varphi )\, \widetilde{\kappa }\mathrm{d}y. \end{aligned}$$

By Lemma D.8, we have

$$\begin{aligned} ||\sigma D_t^{k}(D_t^2\varphi ) -D_t^k (\sigma D_t^2 \varphi )||_{L^2(\Omega )} \le P(L, ||\varphi ||_{k-1})||\varphi ||_{k}, \end{aligned}$$

and by the commutator estimate (D.12),

$$\begin{aligned} \begin{aligned} ||[D_t^{k+1}, \widetilde{\partial }_j] \varphi ||_{L^2(\Omega )}&\le C_k(M, ||\widetilde{x}||_k, ||V||_{\mathcal {X}^{k}}) \big ( ||\widetilde{\partial }f||_{k, 0} + (||V||_{\mathcal {X}^{k+1}} + 1) ||\widetilde{\partial }f||_{k-1}\big ),\\ ||[D_t^k, \widetilde{\partial }_j] \widetilde{\partial }_i \varphi ||_{L^2(\Omega )}&\le C_k(M, ||\widetilde{x}||_k, ||V||_{\mathcal {X}^k}) (||\widetilde{\partial }^2 f||_{k-1, 0} + ||\widetilde{\partial }^2 f||_{k-2}). \end{aligned} \end{aligned}$$

By (A.25), $||\widetilde{\partial }^{2\!\!} f||_{k-1\!,0}\! \le \! C(M,\! ||V||_k)(1\! +\! ||V\!||_{\mathcal {X}^{k\!+\!1}} )||\widetilde{\partial }f||_{k}$ and since $D_t^k(\sigma D_t^{2} \varphi ) - D_t^k \!\widetilde{\Delta }\varphi \!=\! D_t^k \! F\!$, using (6.4) to control $D_t^kF$, we have (F.37). To prove (F.36) from (F.37), we want to re-write $||\varphi ||_s$ in terms of $||\varphi ||_{s,0}$ and $||\widetilde{\partial }\varphi ||_{s-1}$, and for this we re-write $\partial _a \varphi \!=\! A_{\,\,a}^i\widetilde{\partial }_i \varphi$ and use (A.25) and Lemma D.1 to get: $|| A_{\,\,a}^i\widetilde{\partial }_i \varphi ||_{s-1} \!\le \! C(M)||\widetilde{x}||_{s} ||\widetilde{\partial }\varphi ||_{s-1}.$ This implies that $||\varphi ||_s \!\le \! C_s \big (||\varphi ||_{s,0} \!+ \!||\widetilde{\partial }\varphi ||_{s-1}\!\big ),$ and so inserting this into (F.37), applying (6.11) and bounding $||V||_s \!\le \! ||V||_{\mathcal {X}^{s+1}}$ and using that ${d}W_{\!s}^2\!/{dt} \!= \! 2 W_{\!s} {d}W_{\!s}/{dt}$ gives (F.36). $\quad \square$

Lemma F.9

There is a continuous function $G_s'' \!=\! G_s''(M,{}_{\!} ||\widetilde{x}||_s)$ and $P_{\!s}$ so that if (6.1)–(6.6) hold, then

$$\begin{aligned}&||\widetilde{\partial }\varphi ||_s \le G_s''(||{\mathcal {T}^{}}\widetilde{x}||_{H^s} + ||V||_{s}) \big ( ||\varphi ||_{s+1,0} + ||\widetilde{\partial }\varphi ||_{s,0} \nonumber \\&\quad + ||F||_{s-1} + P_s(L,||\varphi ||_{s,0}, ||\widetilde{\partial }\varphi ||_{s-1,0}, ||F||_{s-2})\big ). \end{aligned}$$

(F.38)

Proof

For $s = 0$ there is nothing to prove and so we assume that (F.38) holds for $s = 0, 1,..., n-1$. To prove that this holds for $s = n$, we will show that if $k + \ell = n$, then

$$\begin{aligned} ||D_t^k \widetilde{\partial }\varphi ||_{H^{\ell }} \le G''_{\!n} \big ( ||\varphi ||_{n+1,0} + ||\widetilde{\partial }\varphi ||_{n,0} + ||F||_{n-1} + P(L, ||\varphi ||_{n,0}, ||\widetilde{\partial }\varphi ||_{n-1,0}, ||F||_{n-2})\big ), \nonumber \\ \end{aligned}$$

(F.39)

with $G_n'' = G_n''(M, ||\widetilde{x}||_n)$. There is nothing to prove that $\ell = 0$ and so we assume that this estimate holds for $\ell = 0, ..., \ell '-1$. To prove that it holds for $\ell = \ell '$, we use the estimate (5.8) when $\ell ' = n$:

$$\begin{aligned}&||\widetilde{\partial }\varphi ||_{{}_{\!}H^{n{}_{\!}}} \!\le \!C'_{\!n} {}_{\!}\big ( ||\Delta \varphi ||_{{}_{\!}H^{n\!-\!1}}\! + (||{\mathcal {T}^{}}{}_{\!} \widetilde{x}||_{{}_{\!}H^n}\! + ||\widetilde{x}||_{{}_{\!}H^n}\!)||\varphi ||_{{}_{\!}L^2}\!\big ) \\&\quad \!\le \! C'_{\!n} {}_{\!}\big ( ||\sigma \!D_t^2 \varphi ||_{{}_{\!}H^{\ell '\!\!-\!1}}\! + ||F||_{{}_{\!}H^{\ell '\!\!-\!1}}\! + (||{\mathcal {T}^{}}{}_{\!} \widetilde{x}||_{{}_{\!}H^n}\! + ||\widetilde{x}||_{{}_{\!}H^n}\!)||\varphi ||_{{}_{\!}L^2}\!\big ), \end{aligned}$$

and the estimate (5.9) when $n-\ell ' \ge 1$:

$$\begin{aligned}&||D_t^{n-\ell '}\! \widetilde{\partial }\varphi ||_{H^{\ell '{}_{\!}}\!(\Omega )} \!\le C'_n \big ( ||\Delta \varphi ||_{n-\ell '\!, \ell '\!-1}\! + (||D_t \widetilde{x}||_n + ||\widetilde{x}||_{n})||D_t^{n-\ell '}\!\! \varphi ||_{L^2(\Omega )}\big )\\&\quad \le C'_n \big ( ||\sigma D_t^2 \varphi ||_{n-\ell '\!, \ell '\!-1}\! + ||F||_{n-\ell '\!, \ell '\!-1} + (||D_t \widetilde{x}||_n + ||\widetilde{x}||_{n})||D_t^{n-\ell '}\!\! \varphi ||_{L^2(\Omega )}\big ). \end{aligned}$$

Using (D.16), the first term here is bounded by $C ||\varphi ||_{n-\ell '+2, \ell '-1} + P(L, ||\varphi ||_{n-1})$ and this second term can be bounded by the right-hand side of (F.39) by the inductive assumption. If $\ell ' = 1$ then we have just proven (F.39). If $\ell ' \ge 2$ we write $\partial \varphi /\partial {y^a} = A_{\,\,a}^i \widetilde{\partial }_i \varphi$ and use the product estimate (A.25) and Lemma D.1 to get

$$\begin{aligned}&||\varphi ||_{n-\ell '+2, \ell '-1} \le ||\partial _y \varphi ||_{n-\ell '+2, \ell '-2} + ||\varphi ||_{n-\ell '+2, \ell '-2} \\&\quad \le C(M, ||\widetilde{x}||_n) (||\widetilde{\partial }\varphi ||_{n-\ell '+2, \ell '-2} + ||\varphi ||_{n-1}); \end{aligned}$$

noting that $||\widetilde{x}||_n \le C (||\widetilde{x}||_{H^n} + ||V||_{n-1})$, this implies (F.39). $\quad \square$

Proof

We will show that

$$\begin{aligned}&\frac{d}{dt\!\!}\, W_s^{{}_{\!}I\!,{}_{\!}I{\!}I} \!\!\le \! D_{\!{}_{\!}s}' \!\bigg (\! ||F_{\!I} \!-\! F_{{}_{\!}I{\!}I\!}||_{s,0} \!+_{\!} ||F_{\!I} \!- \!F_{{}_{\!}I{\!}I\!}||_{s-\!1}\! +_{\!} W_s^{{}_{\!}I\!,{}_{\!}I{\!}I} \!\! \nonumber \\&\quad +_{\!} ||V_{{}_{\!}{}_{\!}I}\!-\! V_{{}_{\!}I{\!}I}\!||_{\mathcal {X}^{{}_{\!}s+_{\!}1\!}} \big (||\varphi _{{}_{\!}I{\!}I\!}||_{s+_{\!}2,0}\! +_{\!} ||\widetilde{\partial }_{{}_{\!}I{\!}I}\varphi _{{}_{\!}I{\!}I\!}||_{s, 1}\! +_{\!} ||\varphi _{{}_{\!}I{\!}I\!}||_{s+_{\!}1_{\!},0} \!+_{\!} ||\widetilde{\partial }_{{}_{\!}I{\!}I}\varphi _{{}_{\!}I{\!}I\!}||_{s\!} \big )\! \bigg )_{\!}, \nonumber \\ \end{aligned}$$

(F.40)

where $D_{\!s}'$ depends on $M, L, W_s^{I_{\!},{}_{\!}I{\!}I\!}(t)$ and $||\widetilde{\partial }_J \varphi _{J\!}||_s, ||\varphi _{J\!}||_{s+1,0}$ for $J_{\!} \!=\! I_{\!},{}_{\!}I{\!}I{}_{\!}$. Arguing as in the proof of Theorem 6.1 and using (6.7) and (6.8), this implies (6.16). Using (6.14) and (), it just remains to prove that the $L^2$ norms of $D_t^s (\widetilde{\Delta }_{{}_{\!}I}\!- \!\widetilde{\Delta }_{{}_{\!}I{\!}I}) \varphi _{{}_{\!}I{\!}I}$ and $D_t^s ( (\sigma _I \!- \sigma _{{}_{\!}I{\!}I}) D_t^2 \varphi _{{}_{\!}I{\!}I})$ are bounded by (F.40). These terms are the reason that we lose derivatives of $\varphi _{{}_{\!}I{\!}I}$ relative to $\psi$ and why the coefficients $D_{\!s}$ will depend on $||V_{{}_{\!}{}_{\!}I}||_{\mathcal {X}^{s+2}}, ||V_{{}_{\!}I{\!}I}||_{\mathcal {X}^{s+2}}$.

We start by controlling $D_t^s (\widetilde{\Delta }_{{}_{\!}I}- \widetilde{\Delta }_{{}_{\!}I{\!}I})\varphi _{{}_{\!}I{\!}I}$ in $L^2$. We write

$$\begin{aligned} D_t^s \big ( \widetilde{\partial }_{I i} \widetilde{\partial }_{I j} - \widetilde{\partial }_{{}_{\!}I{\!}Ii}\widetilde{\partial }_{{}_{\!}I{\!}Ii}\big ) \varphi _{{}_{\!}I{\!}I}&= \big ( \widetilde{\partial }_{I i} D_t^s\widetilde{\partial }_{I j} - \widetilde{\partial }_{{}_{\!}I{\!}Ii}D_t^s\widetilde{\partial }_{{}_{\!}I{\!}Ii}\big ) \varphi _{{}_{\!}I{\!}I}\\&\quad + \big ( [\widetilde{\partial }_{I i},D_t^s ]\widetilde{\partial }_{I j} - [\widetilde{\partial }_{{}_{\!}I{\!}Ii},D_t^s]\widetilde{\partial }_{{}_{\!}I{\!}Ii}\big ) \varphi _{{}_{\!}I{\!}I}. \end{aligned}$$

The first term is bounded in $L^2(\Omega )$ by $C(M) ||(\widetilde{\partial }_{{}_{\!}I}\!- \widetilde{\partial }_{{}_{\!}I{\!}I}{}_{\!})\varphi _{II}||_{s,1}$. By the product rule (A.25) this term is bounded by $C(M{}_{\!}, {}_{\!}||V_{{}_{\!}{}_{\!}I}||_{\mathcal {X}^{s+1}}, ||V_{{}_{\!}I{\!}I}||_{\mathcal {X}^{s+1}}{}_{\!})||V_{{}_{\!}{}_{\!}I}\!- \!V_{{}_{\!}I{\!}I}||_{\mathcal {X}^{s+1}}||\widetilde{\partial }_{{}_{\!}I{\!}I}\varphi _{{}_{\!}I{\!}I}||_{s+1}.$ Using the commutator estimate (D.9), the $L^2$ norm of the second term is bounded by the right-hand side of (F.40).

To control $D_t^s (\sigma _I - \sigma _{{}_{\!}I{\!}I}) D_t^2 \varphi _{{}_{\!}I{\!}I}$, we use (A.25) to get

$$\begin{aligned} ||D_t^s ( (\sigma _I - \sigma _{II} )D_t^2 \varphi _{{}_{\!}I{\!}I})||_{L^2(\Omega )} \le D_s'' ||\varphi _I - \varphi _{{}_{\!}I{\!}I}||_{s,0} ||\varphi _{{}_{\!}I{\!}I}||_{s+2,0}, \end{aligned}$$

where $D_s''$ depends on L,  and $||\varphi _J||_{s-1}$ for $J = I, {}_{\!}I{\!}I$.

The estimate (6.17) follows in the same way as for (6.8) using the elliptic estimate (B.5) in place of (5.9) and

$$\begin{aligned} \widetilde{\Delta }_{{}_{\!}I}\varphi _I -\widetilde{\Delta }_{{}_{\!}I{\!}I}\varphi _{{}_{\!}I{\!}I} = \sigma _I D_t^2 \psi + (\sigma _I - \sigma _{{}_{\!}I{\!}I}) D_t^2 \varphi _{{}_{\!}I{\!}I} +F_I - F_{{}_{\!}I{\!}I}. \end{aligned}$$

$\square$