On the local well-posedness for the relativistic Euler equations for a liquid body

Abstract

We prove a local existence theorem for the free boundary problem for a relativistic fluid in a fixed spacetime. Our proof involves an a priori estimate which only requires control of derivatives tangential to the boundary, which holds also in the Newtonian compressible case.

Appendices

Appendix A. Tangential smoothing, fractional derivatives, vector fields and norms

1.1 A.0.1. The tangential derivatives and tangential norms

Since $\Omega$ is the unit ball, the vector fields

$$\begin{aligned} \Omega _{ab} = y^a \partial _{y^b} - y^b \partial _{y^a} , \qquad a,b = 1,2,3, \end{aligned}$$

(A.1)

are tangent to $\partial \Omega$ and span the tangent space there. With $\eta$ the cutoff function defined above, let:

$$\begin{aligned} {\mathcal {S}}= \cup _{a,b \,= 1,2,3}\{ \eta \, \Omega _{ab}, (1-\eta ) \partial _{y^a}\}. \end{aligned}$$

(A.2)

In analogy with the two dimensional case, when ${\mathcal {S}}$ is just the derivative with respect to the angle in polar coordinates, we will now introduce some simplified notation for the norms. Suppose that $f:\Omega \rightarrow {\textbf{R}}$ is a function and ${\mathcal {S}}\!=\!\{S_1,\dots ,S_{N}\}$ is a family of vector fields that are tangential to the boundary at the boundary that span the tangent space there. Let ${\mathcal {S}}f$ stand for the map ${\mathcal {S}}f\!\!:\!\Omega \rightarrow \!{\textbf{R}}^{N}\!\!\!$, whose components are $S_j f$, for $j\!{}_{\!}={}_{\!}\!1,{}_{\!}...,{}_{\!}N\!$. For r an integer, let ${\mathcal {S}}^r\!\!{}_{\!}={}_{\!}\!{\mathcal {S}}\!\!\times \!{}_{\!}\cdots \!\times \! {\mathcal {S}}$(r times) and let $S^I\!\!\in \! {\mathcal {S}}^r\!$ stand for a product of r vector fields in ${\mathcal {S}}\!$, where $I\!=\!(i_1,{}_{\!}...,{}_{\!}i_{r\!})\!\in \! [1,N]\!\times \!\cdots \!\times \! [1,N]$ is a multiindex of length $|I|\!=\!r$. Let ${\mathcal {S}}^r f\!$ stand for the map ${\mathcal {S}}^r\! f\!\!:\!\Omega \!\rightarrow \!{\textbf{R}}^{N r}\!\!\!$, whose components are $S^I \!f$, for $1\!\le \! i_j\!\le \! N\!$, $j\!=\!1,{}_{\!}...{}_{{}_{\!}},r$. The norm of ${\mathcal {S}}^r\! f\!$ is

$$\begin{aligned} |{\mathcal {S}}^r \!f|^2= {\mathcal {S}}^r \!f\cdot {\mathcal {S}}^r \! f,\quad \text {where}\quad {\mathcal {S}}^r f\cdot {\mathcal {S}}^r g= {\sum }_{|I|=r,\,\,S^I\in {\mathcal {S}}^r} S^I\! f \,\, S^I g. \end{aligned}$$

(A.3)

Moreover, let

$$\begin{aligned} ||W||_{H^{k{}_{\!},r}} ={\sum }_{\ell \le r} ||{\mathcal {S}}^\ell W||_{H^{k}(\Omega )}. \end{aligned}$$

We will use similar notation for space time vector fields tangential to the boundary. Let ${\mathcal {T}}\!=\!{\mathcal {S}}\!\cup \! D_t$, and ${\mathcal {T}}^r\!\! =\!{\mathcal {T}}\!{}_{\!}\times \!\!\cdots \!\!\times \!{\mathcal {T}}$($r{\!}$ times), ${\mathcal {T}}^{r\!,k}\!{}_{\!}=\!{\mathcal {S}}^r {}_{\!}\!\!\times \!{}_{\!} D_{{}_{\!}t}^k$. For $K\!{}_{\!}=\!(_{\!}I{}_{\!},{}_{\!}k{}_{{}_{\!}})$ a multiindex with $|I|\!{}_{\!}=\!r$, we write $T^K\!\!=\!S^I \!D_{{}_{\!}t}^k\!$, $S^I\!\!\in \!{\mathcal {S}}^r$.

1.1.1 A.0.2. Global operators defined in terms of local coordinates

There is a family of open sets $V_{{}_{\!}\mu }$, $\mu \!=\!1,\dots ,N$ that cover $\partial \Omega$ and onto diffeomorphisms $\Phi _\mu \!:\! ( - 1,1)^{2 \!}\!\rightarrow \! V_{{}_{\!}\mu }$. We fix a collection of cutoff functions $\chi _{\mu }{}_{\!}\!:\!\partial \Omega \!\rightarrow \!{\mathbb {R}}$ so that $\chi _\mu ^2$ form a partition of unity subordinate to the cover $\{V_\mu \}_{\mu = 1}^N$, as well as another family of “fattened” cutoff functions ${\widetilde{\chi }}_\mu$ so that the support of ${\widetilde{\chi }}_\mu$ is contained in $V_\mu$ and so that ${\widetilde{\chi }}_\mu \!\equiv \! 1$ on the support of $\chi _\mu$. Recalling that $\Omega$ is the unit ball, we set $W_{\!\mu } \!=\! \{r\omega , r\! \in \! (1/2, 1],\, \omega \! \in \!V_{\!\mu }\}$ for $\mu \! = \! 1,\dots , N$ and let $W_0$ be the ball of radius 3/4 so that the collection $\{W_{\!\mu }\}_{\mu = 0}^N$ covers $\Omega$. Then $y\!=\!\Psi _{\!\mu }({\widehat{z}}) \!= z^{3\,}\omega$, where $\omega \!=\!\Phi _\mu (z)$ and ${\widehat{z}}\!=\!(z,z^3)$, is a diffeomorphism $\Psi \!:\!( - 1,1)^2\! \times \! (1/2,1]\!\rightarrow \!W_{\!\mu }$. Let $\eta \!:\![0,1] \!\rightarrow \! {\mathbb {R}}$ be a bump function so that $\eta (r)\! =\! 1$ when $1/2\! \le \!r \!\le \! 1$ and $\eta (r) \!= \!0$ when $r\! < \!1/4$. We define cutoff functions on $\Omega$ by setting $\chi _\mu ({\widehat{z}})\!= \!\chi _\mu (z) \eta (z^3) ,$ for $\mu \!\ge \! 1$, and $\chi _0$ so $\sum \chi _\mu ^2\!=\!1$. Let $\Psi ^\prime _{\!\mu }\!=\! \partial y/\partial {\widehat{z}}$ and $\Psi ^\prime _{\!\mu }\!=\!\partial \omega /\partial z$. Then $\det {\Psi ^\prime _{\!\mu }}\!=\! r^2 \det {\Phi ^\prime _\mu }$.

In the local coordinates the tangential vector fields (A.1) takes the form

$$\begin{aligned} S =S^a(z)\, \partial /\partial z^a, \qquad \text {with}\quad S^3(z)=0. \end{aligned}$$

Moreover we can write

$$\begin{aligned} {\widetilde{\partial }}_i= & {} {\widehat{J}}_i^d {\widehat{\partial }}_d, \quad \text {where}\quad {\widehat{J}}_i^d =\partial {\widehat{z}}^d\!/\partial {\widetilde{x}}^i,\quad \text {and}\quad \\ {\widehat{\partial }}_d= & {} \partial /\partial {\widehat{z}}^d=( \Psi ^\prime _\mu )_d^a\partial _a,\quad \partial _a=\partial /\partial y^a. \end{aligned}$$

For a linear operator A defined in local coordinates on the sphere we define a global operator A by

$$\begin{aligned} Af= & {} {{\,\mathrm{{\textstyle {\sum }}}\,}}A_\mu f,\quad \text {where}\quad A_\mu f\! = \chi _\mu m_\mu ^{-1} {A}\big [m_\mu f_{\!\mu }\big ]_{\!}\circ _{\!}\Psi _{\!\mu }^{-1}\!\!,\qquad \nonumber \\ f_{\!\mu }(z)= & {} (\chi _\mu f)_{\!}\circ _{\!} \Psi _{\!\mu }(z,z^3). \end{aligned}$$

(A.4)

Here $m_\mu =|\det {\Psi _{\!\mu }^\prime }|^{1/2}$ is inserted so that A is symmetric with the measure dy if it is with the measure dz for fixed $z_3$ since $dS(\omega )=m_\mu ^2 dz$. For the smoothing the symmetry in spherical coordinates makes things simpler since it will mean that the global operator defined by (A.4) is symmetric on the sphere.

However for the fractional derivative in only defined locally in each coordinate system so in that case we will pick $m_\mu =1$. Then we have

$$\begin{aligned} {\widehat{\partial }}_d\big ( A[f_\mu ]\circ \Psi ^{-1}\big )= & {} ({\widehat{\partial }}_d A[f_\mu ])\circ \Psi ^{-1} \\= & {} [{\widehat{\partial }}_d, A][f_\mu ]\circ \Psi ^{-1}\!+A[{\widehat{\partial }}_d f_\mu ]\circ \Psi ^{-1\!\!},\qquad \\{} & {} {\widehat{\partial }}_d f_\mu \!=\!( {\widehat{\partial }}_d f)_\mu \!+\big ( {\widehat{\partial }}_d\chi _\mu \, f\big )_{\!}\circ _{\!} \Psi _{\!\mu } \end{aligned}$$

and

$$\begin{aligned} S\big ( A[f_\mu ]\circ \Psi ^{-1}\big )= & {} (S A[f_\mu ])\circ \Psi ^{-1} \\= & {} [S, A][f_\mu ]\circ \Psi ^{-1}+A[S f_\mu ]\circ \Psi ^{-1}\!,\qquad \\{} & {} S f_\mu =( S f)_\mu + \big ( S\chi _\mu \, f\big )_{\!}\circ _{\!} \Psi _{\!\mu } \end{aligned}$$

Hence the commutators between the global operator A and ${\widehat{\partial }}_i$ or S consist of the commutators between these in the local coordinates plus terms when the derivatives fall on the cutoffs or measures which are lower order.

1.1.2 A.0.3. Tangential smoothing

Let $\varphi \!:\! {\mathbb {R}}^2 \!\!\rightarrow \! {\mathbb {R}}$ be even, supported in $R = (-1,1)^2$ with $\int _{{\mathbb {R}}^2} \!\varphi =\! 1$ and

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

be a smoothing operator. Because $\varphi$ is even, $S_\varepsilon$ is symmetric; for any functions $f, g: {\mathbb {R}}^2 \rightarrow {\mathbb {R}}$:

$$\begin{aligned} \int _{{\mathbb {R}}^2} S_\varepsilon f(z)\,\, g(z)\, dz = \int _{{\mathbb {R}}^2} f(z) \,\, S_\varepsilon g(z)\, dz. \end{aligned}$$

We now define global symmetric operators on $\Omega$ or $\partial \Omega$ by (A.4):

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

(A.5)

1.1.3 A.0.4. Commutators with smoothing

We have

Lemma A.1

With $S_\varepsilon$ defined by (A.5), if $k \ge m$ then:

$$\begin{aligned} ||S_\varepsilon f||_{H^k(\partial \Omega )} \lesssim \varepsilon ^{m-k} ||f||_{H^{m}(\partial \Omega )}, \quad \text {and}\quad ||S_\varepsilon f - f||_{H^k(\partial \Omega )} \lesssim \varepsilon ||f||_{H^{k+1}(\partial \Omega )}, \end{aligned}$$

and

$$\begin{aligned} \Vert S_\varepsilon f\Vert _{L^\infty (\partial \Omega )}\le \Vert f\Vert _{L^\infty (\partial \Omega )}. \end{aligned}$$

Moreover, for $k=0,1$:

$$\begin{aligned} ||S_\varepsilon (fg) -f S_\varepsilon g||_{H^k(\partial \Omega )}\lesssim \varepsilon ^{1-k} ||f||_{C^{1+k}(\partial \Omega )} ||g||_{L^2(\partial \Omega )}, \end{aligned}$$

and for $n=0,1$

$$\begin{aligned} ||S_\varepsilon (fg) -f S_\varepsilon g||_{H^{n,k}(\Omega )}\lesssim \varepsilon ^{1-k} ||f||_{C^{n,1+k}(\Omega )} ||g||_{H^n(\Omega )}, \end{aligned}$$

(A.6)

where

$$\begin{aligned} \Vert f\Vert _{C^{n,k}}={\sum }_{|I|\le k,\, S\in {\mathcal {S}}}\Vert S^I\! f\Vert _{C^n}, \quad \text {and}\quad \Vert f\Vert _{H^{n,k}}={\sum }_{|I|\le k,\, S\in {\mathcal {S}}}\Vert S^I\! f\Vert _{H^n}. \end{aligned}$$

Proof

The proof for $k=0$ follows from the local expression and the fact that $|w| \le \varepsilon$ in the support of $\varphi _\varepsilon$,

$$\begin{aligned} S_\varepsilon (fg)(z) - f(z)S_\varepsilon (g)(z) = \int _{{\mathbb {R}}^2} \varphi _\varepsilon (w)g(z-w)\big ( f(z-w) - f(z)\big ) dw. \end{aligned}$$

The proof for $k=1$ follows from differentiating this and integrating by parts if the derivative falls on g, see the proof of Lemma A.2. $\square$

There is an improvement in the commutators with smoothing for tangential derivatives:

Lemma A.2

We have $[S_\varepsilon ,D_t]\!=\!0$. If $S\!=\!S^a(y)\partial _a$ is a tangential vector field then for $k\!=\!0,1$:

$$\begin{aligned} \Vert [S_\varepsilon ,S]g\,\Vert _{H^k(\partial \Omega )}+\Vert [S_\varepsilon ,\partial _r]\,g\Vert _{H^k(\partial \Omega )}&\lesssim \Vert g\Vert _{H^k(\partial \Omega )} , \end{aligned}$$

(A.7)

$$\begin{aligned} ||S_\varepsilon (f S g) - f S_\varepsilon S g||_{H^k(\partial \Omega )}&\lesssim || f||_{C^{k}(\partial \Omega )} ||g||_{H^{k}(\partial \Omega )}. \end{aligned}$$

(A.8)

Moreover for $n=0,1$

$$\begin{aligned} \Vert [S_\varepsilon ,S]g\Vert _{H^{n,k}(\Omega )}+\Vert [S_\varepsilon ,\partial _r]g\Vert _{H^{n,k}(\Omega )}&\lesssim \Vert g\Vert _{H^{n,k}(\Omega )} , \\ ||S_\varepsilon (f S g) - f S_\varepsilon S g||_{H^{n,k}(\Omega )}&\lesssim || f||_{C^{n,k}(\Omega )} ||g||_{H^{n,k}(\Omega )}. \end{aligned}$$

Proof of Lemma A.2

In local coordinates such that $S=S^d(z)\partial /\partial z^d$, with $S^3=0$, we have, neglecting that the measure depends on the coordinates,

$$\begin{aligned} \big ( S_\varepsilon ( S g) - SS_\varepsilon \,g \big )(z) = \int _{{\mathbb {R}}^2} \big ( S^d(z - \varepsilon w) - S^d(z) \big ) \frac{\partial g(z - \varepsilon w)}{\partial z^{d}} \varphi (w) \, dw. \end{aligned}$$

Writing $(S g)(z - \varepsilon w)= S^d(z - \varepsilon w)\varepsilon ^{-1} \partial g(z - \varepsilon w)/\partial w^{d}$ and integrating by parts this becomes:

$$\begin{aligned} \big ( S_\varepsilon ( S g) - SS_\varepsilon \,g \big )(z)= & {} \int _{{\mathbb {R}}^2} \!\! \frac{\partial S^d(z \! -\! \varepsilon w) }{\partial z^d} g(z\!-\!\varepsilon w) \varphi (w) \, dw\\{} & {} + \int _{{\mathbb {R}}^2}\!\! \frac{ S^d(z\! -\! \varepsilon w)\! -\! S^d(z)}{\varepsilon } g(z\!-\!\varepsilon w) \frac{\partial \varphi (w)}{\partial w^{ d}} \, dw. \end{aligned}$$

Both terms are bounded by the right-hand side of (A.7), for $k=0$ and the case $k=1$ follows from differentiating this. In a similar way we have

$$\begin{aligned} \big ( S_\varepsilon ( f S g) - f S_\varepsilon S g \big )(z) = \int _{{\mathbb {R}}^2} \big ( f(z - \varepsilon w) - f(z) \big ) (S g)(z - \varepsilon w) \varphi (w) \, dw, \end{aligned}$$

and integrating by parts as above we get

$$\begin{aligned}{} & {} \big ( S_\varepsilon ( f S g) -{}_{\!} f S_\varepsilon S g \big )(z)\\{} & {} \quad =\! \int _{{\mathbb {R}}^2} \!\!\! (Sf)(z \! - \! \varepsilon w) g(z \!- \!\varepsilon w) \varphi (w) \, dw \\{} & {} \qquad +\int _{{\mathbb {R}}^2}\!\!\!\frac{ f(z\! - \! \varepsilon w)\! -{}_{\!}\! f(z)}{\varepsilon } g(z \!- \!\varepsilon w) \frac{\partial \big ( S^d(z\!- \!\varepsilon w)\varphi (w)\big )\!\!}{\partial w^{ d}} \, dw. \end{aligned}$$

(A.8) follows from this. $\square$

In order to control the commutators $[{\widetilde{\partial }}, S_\varepsilon ]$, we need the following two lemmas:

Lemma A.3

Suppose that

$$\begin{aligned} |\partial {\widetilde{x}}/\partial y|+|\partial y/\partial {\widetilde{x}}|\le M_0. \end{aligned}$$

Then if $S=S^a(y)\partial _a$ is a tangential vector field we have

$$\begin{aligned}{} & {} ||[{\widetilde{\partial }}_i, S_\varepsilon ] S g||_{L^2(\Omega )} + ||S[{\widetilde{\partial }}_i, S_\varepsilon ] g||_{L^2(\Omega )} + ||[{\widetilde{\partial }}_i,S] g||_{L^2(\Omega )} \\{} & {} \quad \lesssim C(M_0) {\sum } _{|I|\le 1} ||\partial S^I {\widetilde{x}} ||_{C^0} ||g||_{H^{1}(\Omega )}. \end{aligned}$$

Proof of Lemma A.3

In the local coordinates such that $S\!=S^d(z)\partial /\partial z^d$, with $S^3\!=0$, we write ${\widetilde{\partial }}_i\!={\widehat{J}}_i^d{\widehat{\partial }}_d$, where ${\widehat{J}}_i^d=\partial {\widehat{z}}^d\!/\partial {\widetilde{x}}^i$, and ${\widehat{\partial }}_d=\partial /\partial {\widehat{z}}^d$. We have $[{\widehat{J}}_i^d{\widehat{\partial }}_d ,S_\varepsilon ]=[{\widehat{J}}_i^d ,S_\varepsilon ]{\widehat{\partial }}_d+{\widehat{J}}_i^d[{\widehat{\partial }}_d ,S_\varepsilon ]$ and

$$\begin{aligned}{}[{\widehat{J}}_i^d{\widehat{\partial }}_d ,S_\varepsilon ]S g=[{\widehat{J}}_i^d ,S_\varepsilon ]S{\widehat{\partial }}_d g+[{\widehat{J}}_i^d ,S_\varepsilon ][{\widehat{\partial }}_d,S]g+{\widehat{J}}_i^d[{\widehat{\partial }}_d ,S_\varepsilon ]S g. \end{aligned}$$

Here the first and main term on the right is dealt with using (A.8). The second one is lower order. The last one is dealt with using (A.7) for $d=1,2$ and the fact that $[{\widehat{\partial }}_d,S_\varepsilon ]=0$. $\square$

1.1.4 A.0.5. The tangential fractional derivatives and norms

We will need to use fractional tangential derivatives to control our solution and we will define these operators in coordinates. If $F: {\mathbb {R}}^2 \rightarrow {\mathbb {R}}$, we define:

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

and we define fractional tangential derivatives on $\Omega$ by:

$$\begin{aligned} \langle \partial _{\theta {}_{\!}} \rangle _\mu ^{s} f = {\widetilde{\chi }}_\mu (\langle \partial _{\theta {}_{\!}} \rangle ^{s} f_\mu )\circ \Psi _{\mu }^{-1},\quad f_\mu =(\chi _\mu f)\circ \Psi , \qquad \mu = 1,..., N. \end{aligned}$$

(A.9)

We also set $\langle \partial _{\theta {}_{\!}} \rangle _0^s f \!=\!\chi _0( \langle \partial \rangle ^{s} f_0)\!\circ \!\Psi _{0}^{-1}\!\!$, where $\langle \partial \rangle ^{s}$ is defined by taking the Fourier transform in all directions.

For $s \in {\mathbb {R}}$, $k \in {\mathbb {N}}$, we define:

$$\begin{aligned} || f ||_{H^s(\partial \Omega )} = {\sum }_{\mu = 1}^N ||\langle \partial _{\theta {}_{\!}} \rangle _\mu ^s f ||_{L^2(\partial \Omega )}, \quad \text {and}\quad || f ||_{H^{(n, s)}(\Omega )} = {\sum }_{\mu = 0}^N ||\langle \partial _{\theta {}_{\!}} \rangle _\mu ^s f||_{H^n(\Omega )}. \end{aligned}$$

For $0<s<1$ let ${\mathcal {S}}^s f:\Omega \rightarrow {\textbf{R}}^N$, or $\langle \partial _\theta \rangle ^s$ be the map whose components are $\langle \partial _{\theta {}_{\!}} \rangle ^{s}_\mu f$, for $\mu =0,\dots ,N$, and define the inner product

$$\begin{aligned} \big (\langle \partial _{\theta {}_{\!}} \rangle ^s f\big ) \cdot \big (\langle \partial _{\theta {}_{\!}} \rangle ^s g\big )={\sum }_{\mu =1,\dots ,N} \big (\langle \partial _{\theta {}_{\!}} \rangle ^s_\mu f\big ) \big ( \langle \partial _{\theta {}_{\!}} \rangle ^s_\mu g\big ). \end{aligned}$$

(A.10)

Moreover let ${\mathcal {S}}^{r+s} f:\Omega \rightarrow {\textbf{R}}^{N+1}$ be the map whose components are $\langle \partial _{\theta {}_{\!}} \rangle ^{s}_\mu S^I f$. The norm of ${\mathcal {S}}^r f\!$ is

$$\begin{aligned} |{\mathcal {S}}^{r+s}\! f|^2= & {} {\mathcal {S}}^{r+s}\! f\cdot {\mathcal {S}}^{r+s}\! f,\quad \text {where}\quad \nonumber \\ {\mathcal {S}}^{r+s}\! f\cdot {\mathcal {S}}^{r+s} g= & {} {\sum }_{\mu =1,\dots ,N}{\sum }_{|I|=r,\,\,S^I\in {\mathcal {S}}^r}\langle \partial _{\theta {}_{\!}} \rangle ^{s}_\mu S^I \!f \, \,\langle \partial _{\theta {}_{\!}} \rangle ^{s}_\mu S^I g. \end{aligned}$$

Lemma A.4

If $S \in {\mathcal {S}}$, then:

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

1.1.5 A.0.6. Commutators with the fractional derivative

In local coordinates we have “Leibniz rule”:

Lemma A.5

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

$$\begin{aligned} ||\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}(FG) - F \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}G||_{L^2({\mathbb {R}}^2)}&\lesssim ||F||_{H^2({\mathbb {R}}^2)} ||G||_{L^2({\mathbb {R}}^2)},\\ ||\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}(FG) - F \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}G||_{H^s({\mathbb {R}}^2)}&\lesssim ||F||_{H^{3}({\mathbb {R}}^2)} ||G||_{H^{s-1/2}({\mathbb {R}}^2)},\qquad 0\le s\le 1, \end{aligned}$$

Proof

The Fourier transform of $\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}(FG) - F \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}G$ is

$$\begin{aligned} \langle \xi \rangle ^{\!1{}_{\!}/2} {\widehat{FG}} (\xi ) - \widehat{ (F \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}G)}(\xi )=\int \big (\langle \xi \rangle ^{\!1{}_{\!}/2} -\langle \xi -\eta \rangle ^{\!1{}_{\!}/2}\big ) {\widehat{F}}(\eta ) {\widehat{G}} (\xi -\eta ) \, d\eta . \end{aligned}$$

Using the elementary estimate $|\langle \xi \rangle ^{1/2} - \langle \xi -\eta \rangle ^{1/2}| \lesssim \langle \eta \rangle \langle \xi \rangle ^{-1/2}$ and Cauchy-Schwarz we have:

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

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

The first estimate now follows from Plancherel’s theorem.

If $s \!\le \!1/2$ we can further estimate $|\langle \xi \rangle ^{1/2} \!- \!\langle \xi \!- \!\eta \rangle ^{1/2}| \langle \xi \rangle ^s \!\lesssim \! \langle \eta \rangle \langle \xi \rangle ^{s-1/2} \!\lesssim \! \langle \eta \rangle ^{3/2-s}\langle \xi \!- \!\eta \rangle ^{s-1/2}$ and if $s\ge 1/2$ we can estimate $|\langle \xi \rangle ^{1/2} \!-\langle \xi \!- \!\eta \rangle ^{1/2}| \langle \xi \rangle ^s\!\lesssim \! \big (\langle \eta \rangle ^{s-1/2} \!+ \langle \xi \!- \!\eta \rangle ^{s-1/2}\big )\langle \eta \rangle$, and this leads to the second estimate. $\square$

We note that our Sobolev norms are independent of change of coordinates:

Lemma A.6

Let $F:{\mathbb {R}}^2 \rightarrow {\mathbb {R}}$ has compact support and let $G=F\circ \Psi$ where $\Psi$ be a $C^1$ diffeomorphism. Then $\Vert \langle \partial _{\theta {}_{\!}} \rangle ^s F\Vert _{L^2({\mathbb {R}}^2)}\lesssim \Vert \langle \partial _{\theta {}_{\!}} \rangle ^s G\Vert _{L^2({\mathbb {R}}^2)}\lesssim \Vert \langle \partial _{\theta {}_{\!}} \rangle ^s F\Vert _{L^2({\mathbb {R}}^2)}$.

Proof

This is directly by changing variables on the space side seen to be true for the $L^2$ part of the norms so it suffices to prove the inequalities for homogeneous Sobolev spaces, i.e. with $\langle \partial _{\theta {}_{\!}} \rangle ^s$ replaced by $|\partial _\theta |^s$. The proof will use the alternative characterization of the fractional Sobolev norms (see Proposition 3.4 in [10]):

$$\begin{aligned} \int \int \frac{|F(x)-F(y)|^2}{|x-y|^{2+2s}} dx dy= C_s \int |\xi |^{2s} |{\widehat{F}}(\xi )|^2 \, d\xi . \end{aligned}$$

With this alternative characterization the proof of the lemma just follows from changing variables, since $|x-y|\lesssim |\Psi (x)-\Psi (y)|\lesssim |x-y|$. $\square$

Lemma A.7

We have

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

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 Lemma A.5 that

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

$\square$

Lemma A.8

For $k=0,1$ we have

$$\begin{aligned} \Vert [\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu \!,{\widehat{\partial }}_d\, ]\, g\Vert _{H^k(\partial \Omega )}&\lesssim \Vert g\Vert _{H^{k+1/2}(\partial \Omega )},\\ \Vert [\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu \!,S\, ]\, g\Vert _{H^k(\partial \Omega )}&\lesssim \Vert g\Vert _{H^{k+1/2}(\partial \Omega )}, \end{aligned}$$

and

$$\begin{aligned} \Vert [\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu \!,{\widehat{\partial }}_d\, ]\, g\Vert _{H^{0,k}(\Omega )}&\lesssim \Vert g\Vert _{H^{0,k+1/2}(\Omega )},\\ \Vert [\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu \!,S\, ]\, g\Vert _{H^{0,k}(\Omega )}&\lesssim \Vert g\Vert _{H^{0,k+1/2}(\Omega )}. \end{aligned}$$

Proof

Since $\langle \partial _{\theta {}_{\!}} \rangle ^{1/2}=\langle ({\widehat{\partial }}_1,{\widehat{\partial }}_2)\rangle ^{1/2}$ commutes with ${\widehat{\partial }}_d$ it is just a matter of ${\widehat{\partial }}_d$ falling on the cutoffs or changes of variables in the definition of $\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu$ which produces a lower order term of the form

$$\begin{aligned} {\widetilde{\chi }}_\mu (\langle \partial _{\theta {}_{\!}} \rangle ^{1/2} g_\mu )\circ \Psi _{\mu }^{-1}\!\!,\qquad g_\mu= & {} \big (({\widehat{\partial }}_d\chi _\mu ) f\big )\circ \Psi _\mu =\sum _\nu \big (({\widehat{\partial }}_d\chi _\mu )\chi _\nu ^2 f \big )\circ \Psi _\nu \circ \Psi _{\nu \mu }\\= & {} \sum _\nu \big (({\widehat{\partial }}_d\chi _\mu )\chi _\nu f_\nu \big ) \circ \Psi _{\nu \mu }, \end{aligned}$$

where $\Psi _{\nu \mu }\!=\!\Psi _\nu ^{-1}\!\circ \Psi _\nu$. The inequalities for ${\widehat{\partial }}_d$ follows directly from Lemma A.5 and Lemma A.6 applied to these. For the case of $S=S^d(z){\widehat{\partial }}_d$ there is an additional commutator in the local coordinates of S and $\langle \partial _{\theta {}_{\!}} \rangle ^{1/2}$ which is also controlled by Lemma A.5. $\square$

As a consequence of the above lemmas we have:

Lemma A.9

We have

$$\begin{aligned} ||\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu (fg) - f \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu g||_{L^2(\partial \Omega )}&\lesssim ||f||_{C^2(\partial \Omega )} ||g||_{L^2(\partial \Omega )},\\ ||\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu (fg) - f \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu g||_{H^1(\partial \Omega )}&\lesssim ||f||_{C^3(\partial \Omega )} || g||_{H^{1/2}(\partial \Omega )},\\ ||\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu (f S g) - f \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu S g||_{L^2(\partial \Omega )}&\lesssim ||f||_{C^3(\partial \Omega )} || g||_{H^{1/2}(\partial \Omega )}. \end{aligned}$$

Moreover, for $n=0,1$

$$\begin{aligned} ||\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu (fg) - f \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu g||_{H^n(\Omega )}&\lesssim ||f||_{C^{n,2}( \Omega )} || g||_{H^n( \Omega )},\nonumber \\ ||\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu (fg) - f \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu g||_{H^{n,1}(\Omega )}&\lesssim ||f||_{C^{n,3}( \Omega )} || g||_{H^{n,1/2}( \Omega )},\nonumber \\ ||\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu (f S g) - f \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu S g||_{H^{n,0}(\Omega )}&\lesssim ||f||_{C^{n,3}( \Omega )} || g||_{H^{n,1/2}( \Omega )}. \end{aligned}$$

(A.11)

Moreover

Lemma A.10

Suppose that

$$\begin{aligned} |\partial {\widetilde{x}}/\partial y|+|\partial y/\partial {\widetilde{x}}|\le M_0. \end{aligned}$$

We have

$$\begin{aligned} ||\,[{\widetilde{\partial }}_i, \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu ] f||_{L^2(\Omega )} \lesssim C(M_0) {\sum } _{|I|\le 2} ||\partial S^I {\widetilde{x}} ||_{C^0} || f||_{H^{1}(\Omega )}. \end{aligned}$$

and

$$\begin{aligned}{} & {} ||\,S [{\widetilde{\partial }}_i, \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu ] f||_{L^2(\Omega )}+||\, [{\widetilde{\partial }}_i, \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu ] S f||_{L^2(\Omega )}\\{} & {} \quad \lesssim C(M_0) {\sum } _{|I|\le 3} ||\partial S^I {\widetilde{x}} ||_{C^0} || f||_{H^{1,1/2}(\Omega )}. \end{aligned}$$

Proof

Writing ${\widetilde{\partial }}_i\!={\widehat{J}}_i^d{\widehat{\partial }}_d$ we have

$$\begin{aligned}{}[{\widehat{J}}_i^d{\widehat{\partial }}_d ,\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu ]={\widehat{J}}_i^d[{\widehat{\partial }}_d ,\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu ]+[{\widehat{J}}_i^d ,\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu ]{\widehat{\partial }}_d, \end{aligned}$$

where the first term is estimated by Lemma A.8 and the second by Lemma A.9. $\square$

1.1.6 A.0.7. Commutators with smoothing and the fractional derivative

Since both smoothing and fractional derivatives are multiplication operators on the Fourier side it follows that they commute in local coordinates and hence

$$\begin{aligned} \Vert \langle \partial _{\theta {}_{\!}} \rangle ^s S_\varepsilon f\Vert _{H^{k}({\mathbb {R}}^2)}\lesssim \Vert \langle \partial _{\theta {}_{\!}} \rangle ^s f\Vert _{H^{k}({\mathbb {R}}^2)}. \end{aligned}$$

Similarly in local coordinates $[{\widehat{\partial }}_d,S_\varepsilon ]$ is either 0 or lower order. Therefore as in the proof of Lemma A.8 we have

Lemma A.11

For $k=0,1$, $0\le s\le 1$

$$\begin{aligned} \Vert \langle \partial _{\theta {}_{\!}} \rangle ^s_\mu S_\varepsilon f\Vert _{H^{k}(\partial \Omega )}&\lesssim \Vert f\Vert _{H^{k+s}(\partial \Omega )},\\ \Vert [\langle \partial _{\theta {}_{\!}} \rangle ^s_\mu , S_\varepsilon ] f\Vert _{H^{k}(\partial \Omega )}&\lesssim \Vert f\Vert _{H^{k+s-1}(\partial \Omega )},\\ \Vert [{\widehat{\partial }}_d , S_\varepsilon ] f\Vert _{H^{k}(\partial \Omega )}&\lesssim \Vert f\Vert _{H^{k-1}(\partial \Omega )}, \end{aligned}$$

and for $n=0,1$

$$\begin{aligned} \Vert \langle \partial _{\theta {}_{\!}} \rangle ^s_\mu S_\varepsilon f\Vert _{H^{n,k}(\Omega )}&\lesssim \Vert f\Vert _{H^{n,k+s}(\Omega )},\\ \Vert [\langle \partial _{\theta {}_{\!}} \rangle ^s_\mu ,S_\varepsilon ] f\Vert _{H^{n,k}(\Omega )}&\lesssim \Vert f\Vert _{H^{n,k+s-1}(\Omega )},\\ \Vert [{\widehat{\partial }}_d ,S_\varepsilon ] f\Vert _{H^{n,k}(\Omega )}&\lesssim \Vert f\Vert _{H^{n,k-1}(\Omega )}. \end{aligned}$$

We can now generalize Lemma A.2 to estimate in the fractional norm:

Lemma A.12

We have $[S_\varepsilon ,D_t]\!=\!0$. If $S\!=\!S^a(y)\partial _a$ is a tangential vector field then for $0\le s\le 1$:

$$\begin{aligned} \Vert [S_\varepsilon ,S]g\,\Vert _{H^s(\partial \Omega )}+\Vert [S_\varepsilon ,\partial _r]\,g\Vert _{H^s(\partial \Omega )}&\lesssim \Vert g\Vert _{H^s(\partial \Omega )} , \end{aligned}$$

(A.12)

$$\begin{aligned} ||S_\varepsilon (f S g) - f S_\varepsilon S g||_{H^s\partial \Omega )}&\lesssim || f||_{C^{1}(\partial \Omega )} ||g||_{H^{s}(\partial \Omega )}. \end{aligned}$$

(A.13)

Moreover for $n=0,1$

$$\begin{aligned} \Vert [S_\varepsilon ,S]g\Vert _{H^{n,s}(\Omega )}+\Vert [S_\varepsilon ,\partial _r]g\Vert _{H^{n,s}(\Omega )}&\lesssim \Vert g\Vert _{H^{n,s}(\Omega )} , \end{aligned}$$

(A.14)

$$\begin{aligned} ||S_\varepsilon (f S g) - f S_\varepsilon S g||_{H^{n,s}(\Omega )}&\lesssim || f||_{C^{n,1}(\Omega )} ||g||_{H^{n,s}(\Omega )}. \end{aligned}$$

(A.15)

Proof

By the proof of Lemma A.2 in local coordinates such that $S=S^d(z)\partial /\partial z^d$, with $S^3=0$, we have, neglecting that the measure depends on the coordinates,

$$\begin{aligned} \big ( S_\varepsilon ( S g) - SS_\varepsilon \,g \big )(z)= & {} \int _{{\mathbb {R}}^2} \!\! \frac{\partial S^d(z \! -\! \varepsilon w) }{\partial z^d} g(z\!-\!\varepsilon w) \varphi (w) \, dw \\{} & {} + \int _{{\mathbb {R}}^2}\!\! \frac{ S^d(z\! -\! \varepsilon w)\! -\! S^d(z)}{\varepsilon } g(z\!-\!\varepsilon w) \frac{\partial \varphi (w)\!}{\partial w^{ d}} \, dw. \end{aligned}$$

Since by Lemma A.6 the fractional Sobolev norm is invariant under changes of coordinates and the same coordinate system works in the overlap of the cutoffs we can apply Lemma A.5 in the same coordinate system as the smoothing to the expression above below the integral signs and that gives (A.12) and (A.14).

Also by the proof of Lemma  A.2

$$\begin{aligned}{} & {} \big ( S_\varepsilon ( f S g) - {}_{\!}f S_\varepsilon S g \big )(z)\\{} & {} \quad =\! \int _{{\mathbb {R}}^2} \!\!\! (Sf)(z \! - \! \varepsilon w) g(z \!- \!\varepsilon w) \varphi (w) \, dw \\{} & {} \qquad +\int _{{\mathbb {R}}^2}\!\!\!\frac{ f(z\! - \! \varepsilon w)\! -{}_{\!}\! f(z)}{\varepsilon } g(z \!- \!\varepsilon w) \frac{\partial \big ( S^d(z\!- \!\varepsilon w)\varphi (w)\big )\!\!}{\partial w^{ d}} \, dw, \end{aligned}$$

and similarly applying Lemma A.5 below the integral signs give (A.13) and (A.15). $\square$

Combining the above lemmas we get:

Lemma A.13

We have

$$\begin{aligned} ||\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu S_\varepsilon (fg) - f \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu S_\varepsilon g||_{L^2(\partial \Omega )}&\lesssim ||f||_{C^2(\partial \Omega )} ||g||_{L^2(\partial \Omega )},\\ ||\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu S_\varepsilon (fg) - f \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu S_\varepsilon g||_{H^1(\partial \Omega )}&\lesssim ||f||_{C^3(\partial \Omega )} || g||_{H^{1/2}(\partial \Omega )},\\ ||\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu S_\varepsilon (f S g) - f \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu S_\varepsilon S g||_{L^2(\partial \Omega )}&\lesssim ||f||_{C^3(\partial \Omega )} || g||_{H^{1/2}(\partial \Omega )}. \end{aligned}$$

Moreover, for $n=0,1$

$$\begin{aligned} ||\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu S_\varepsilon (fg) - f \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu S_\varepsilon g||_{H^n(\Omega )}&\lesssim ||f||_{C^{n,2}( \Omega )} || g||_{H^n( \Omega )},\\ ||\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu S_\varepsilon (fg) - f \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu S_\varepsilon g||_{H^{n,1}(\Omega )}&\lesssim ||f||_{C^{n,3}( \Omega )} || g||_{H^{n,1/2}( \Omega )},\\ ||\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu S_\varepsilon (f S g) - f \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu S_\varepsilon S g||_{H^{n,0}(\Omega )}&\lesssim ||f||_{C^{n,3}( \Omega )} || g||_{H^{n,1/2}( \Omega )}. \end{aligned}$$

Lemma A.14

Suppose that

$$\begin{aligned} |\partial {\widetilde{x}}/\partial y|+|\partial y/\partial {\widetilde{x}}|\le M_0. \end{aligned}$$

We have

$$\begin{aligned} ||\, [{\widetilde{\partial }}_i, \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu S_\varepsilon ] f||_{L^2(\Omega )} \lesssim C(M_0) {\sum } _{|I|\le 2} ||\partial S^I {\widetilde{x}} ||_{C^0} || f||_{H^{1}(\Omega )}, \end{aligned}$$

(A.16)

and

$$\begin{aligned}{} & {} ||\,S [{\widetilde{\partial }}_i,{}_{\!} \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu S_\varepsilon ] f||_{L^2(\Omega )} +||\, [{\widetilde{\partial }}_i,{}_{\!} \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu S_\varepsilon ] S f||_{L^2(\Omega )} \nonumber \\{} & {} \quad \lesssim C(M_0) {\sum } _{|I|\le 3} ||\partial S^I {\widetilde{x}} ||_{C^0} || f||_{H^{1,1/2}(\Omega )}. \end{aligned}$$

(A.17)

Proof

We have $[{\widetilde{\partial }}_i, \! \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu \!S_\varepsilon ] f\!=\![{\widetilde{\partial }}_i, \!\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu ]S_\varepsilon f\!+\!\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu [{\widetilde{\partial }}_i, S_\varepsilon ] f$. The first term is estimated by Lemma A.10 and the second term can be estimated by Lemma A.3. This proves (A.16) and (A.17) for the first term with S to the left of the commutator.

It remains to prove (A.17) for the second term with S to the right of the commutator. We have

$$\begin{aligned}{}[{\widetilde{\partial }}_i, \! \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu \!S_\varepsilon ]S f\!=\![{\widetilde{\partial }}_i, \!\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu ]S_\varepsilon S f\!+\!\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu [{\widetilde{\partial }}_i, S_\varepsilon ]S f. \end{aligned}$$

Here

$$\begin{aligned}{}[{\widetilde{\partial }}_i, \!\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu ]S_\varepsilon S f=[{\widetilde{\partial }}_i, \!\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu ]SS_\varepsilon f\!+\![{\widetilde{\partial }}_i, \!\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu ][S_\varepsilon ,S] f, \end{aligned}$$

where the first term is estimated by Lemma A.10 and Lemma A.11, and the second by Lemma A.2 and Lemma A.10. Hence it remains to estimate

$$\begin{aligned} \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu [{\widetilde{\partial }}_i, S_\varepsilon ]S f=\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu [{\widehat{J}}_i^d{\widehat{\partial }}_d, S_\varepsilon ]S f \!=\!\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu [{\widehat{J}}_i^d \!,S_\varepsilon ]{\widehat{\partial }}_d S f\!+\!\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu \!{\widehat{J}}_i^d[{\widehat{\partial }}_d ,S_\varepsilon ]S f. \end{aligned}$$

Since $[{\widehat{\partial }}_d,S]$ is either 0 or a tangential vector field it follows that $\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu [{\widehat{J}}_i^d \!,S_\varepsilon ][{\widehat{\partial }}_d, S]f$ can be estimated by Lemma A.2. Moreover $[\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu \!,{\widehat{J}}_i^d][{\widehat{\partial }}_d ,S_\varepsilon ]S f$ can be estimated by Lemma A.9 and Lemma A.2. Hence it remains to estimate

$$\begin{aligned} \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu [{\widehat{J}}_i^d \!,S_\varepsilon ]S{\widehat{\partial }}_d f +{\widehat{J}}_i^d\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu [{\widehat{\partial }}_d ,S_\varepsilon ]S f. \end{aligned}$$

Here the $L^2$ norm of the second term can be estimated by the $L^\infty$ norm of ${\widehat{J}}_i^d$ time the $L^2$ norm of the other factor which is a tangential pseudo differential operator of order 3/2 so it is under control by the right hand side of (A.17). Hence it remains to estimate

Here

$$\begin{aligned} \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu {\widehat{J}}_i^d S_\varepsilon S{\widehat{\partial }}_d f ={\widehat{J}}_i^d \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu S_\varepsilon S{\widehat{\partial }}_d f +[\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu \!,{\widehat{J}}_i^d ] SS_\varepsilon {\widehat{\partial }}_d f +[\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu \!,{\widehat{J}}_i^d ][S_\varepsilon , S]{\widehat{\partial }}_d f, \end{aligned}$$

where the second term in the right can be estimated by Lemma A.9 and Lemma A.11 and the third term by Lemma A.9 and Lemma A.2. Hence it remains to estimate

which follows from Lemma A.13. $\square$

Appendix B. Basic elliptic estimates

We collect here some elliptic estimates which will be used in the course of the proof. These estimates all appear in [16] as well as in some of the earlier references [23, 7].

1.1 B.0.1. The estimates used to estimate $V\!\!$

The proof of the following lemma can be found in [16, 23].

Lemma B.1

There is a constant $c_0 = c_0(|\partial {\widetilde{x}}|)$ so that if $\alpha$ is a (0, 1)-tensor on $\Omega$ then

$$\begin{aligned} |{\widetilde{\partial }} \alpha | \le c_0\big ( |{\widetilde{{\text {div}}}} \alpha |+ |{\widetilde{{\text {curl}}}} \alpha | + |{\mathcal {S}}\alpha |\big ), \qquad \text { on } \Omega . \end{aligned}$$

(B.1)

1.1.1 B.0.2. The improved half derivative estimates used to estimate the coordinates

Proposition B.2

There is a constant $C_{\!0}\!$ depending on $\Vert \partial {\widetilde{x}}\Vert _{L^\infty (\Omega )}\!$ so that if $\alpha \!$ is a vector field then

$$\begin{aligned} \! ||\alpha ||_{H^{1{}_{\!}}(\Omega )}^2\!\le & {} \! C_{{}_{\!}0\!}\Big (||{\text {div}}\,\alpha ||_{L^2(\Omega )}^2\! +{}_{\!} ||{\text {curl}}\,\alpha ||_{L^2(\Omega )}^2\! \nonumber \\{} & {} + \! \!\int _{\partial \Omega }\!\!\!\! {\mathcal {N}}_{{}_{\!}i} {\mathcal {N}}_{\!j}\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}\! \alpha ^i \!\cdot _{\!} \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}\! \alpha ^j dS + ||\alpha ||_{L^2(\partial \Omega )}^2\! + ||\alpha ||_{L^2(\Omega )}^2\!\Big ).\!\!\!\! \end{aligned}$$

(B.2)

Here $\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}$ is a half angular derivative defined locally in coordinates in (A.9), and the inner product is the sum over coordinate charts $\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}\alpha ^i \cdot \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}\alpha ^j=\sum _{\mu } \big (\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu \alpha ^i\big ) \big (\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu \alpha ^j\big )$ in (A.10). Moreover

$$\begin{aligned} \! ||\alpha ||_{H^{1{}_{\!}}(\Omega )}^2\!\le & {} \! C_{{}_{\!}1\!}\Big ( ||{\text {div}}\,\alpha ||_{L^2(\Omega )}^2\! +{}_{\!} ||{\text {curl}}\,\alpha ||_{L^2(\Omega )}^2\! \\{} & {} + \! \!\int _{\partial \Omega }\!\!\!\!\gamma _{ij}\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}\! \alpha ^i \!\cdot _{\!} \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}\! \alpha ^j dS + ||\alpha ||_{L^2(\partial \Omega )}^2\! + ||\alpha ||_{L^2(\Omega )}^2\!\Big ). \end{aligned}$$

where $\gamma _{ij}$ is the tangential metric.

Proposition B.3

There is a constant $C_{\!0}\!$ depending on $\Vert \partial {\widetilde{x}}\Vert _{L^\infty (\Omega )}\!$ so that if $\beta \!$ is a vector field then

$$\begin{aligned}{} & {} ||\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}\beta ||_{H^{1{}_{\!}}(\Omega )}^2\!\\{} & {} \quad \le \! C_{{}_{\!}1\!}\Big (||{\text {div}}\,\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}\beta ||_{L^2(\Omega )}^2\! +{}_{\!} || {\text {curl}}\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}\beta ||_{L^2(\Omega )}^2\! \\{} & {} + \! \!\int _{\partial \Omega }\!\!\!\! {\mathcal {N}}_{{}_{\!}i} {\mathcal {N}}_{\!j\,} {\mathcal {S}}\beta ^i \!\cdot {\mathcal {S}}\beta ^j dS + ||\beta ||_{L^2(\partial \Omega )}^2\! + || {\mathcal {S}}\beta ||_{L^2(\Omega )}^2\!\Big ). \end{aligned}$$

Here $\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}$ is a half angular derivative defined locally in coordinates in (A.9), and $S \beta ^i\!\cdot {\mathcal {S}}\beta ^j$ is the inner product of all tangential derivatives defined in (A.3). Moreover

$$\begin{aligned} \! ||\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}\beta ||_{H^{1{}_{\!}}(\Omega )}^2\!\le & {} \! C_{{}_{\!}1\!}\Big ( ||{\text {div}}\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}\beta ||_{L^2(\Omega )}^2\! +{}_{\!} ||{\text {curl}}\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}\beta ||_{L^2(\Omega )}^2\! \\{} & {} + \! \!\int _{\partial \Omega }\!\!\!\!\gamma _{ij\,} {\mathcal {S}}\beta ^i \!\cdot {\mathcal {S}}\beta ^j dS + ||\beta ||_{L^2(\partial \Omega )}^2\! + || {\mathcal {S}}\beta ||_{L^2(\Omega )}^2\!\Big ). \end{aligned}$$

where $\gamma _{ij}$ is the tangential metric.

These propositions are a consequence of the following two lemmas proven below:

Lemma B.4

If $\alpha$ is a vector field then:

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

Lemma B.5

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

$$\begin{aligned} \Big |\int _{\partial \Omega } \big (\gamma ^{ij} - {\mathcal {N}}^i{\mathcal {N}}^j\big )\alpha _i \alpha _j \kappa dy_\gamma \Big |{} & {} \le 2\Big |\int _{\Omega } {\text {div}}( \alpha )\, \alpha _{j}{\mathcal {N}}^j+ {\text {curl}}\alpha _{ij}\, \alpha ^i {\mathcal {N}}^j dx \Big | \\{} & {} \quad + K ||\alpha ||_{L^2(\Omega )}^2. \end{aligned}$$

We also need estimates for the Dirichlet problem that keep track of the regularity of the boundary and that uses the minimal amount of regularity of the boundary:

Proposition B.6

Suppose that $q=0$ on $\partial \Omega$. Then

and

where $c_K$ depends on ${\widetilde{\partial }} T^{N}{\widetilde{x}}$ and ${\widetilde{\partial }} T^{N}{\widetilde{\partial }}q$, for $|N|\le |K|/2+3$.

The proof follows from Lemma B.7 below

Lemma B.7

Suppose that $q=0$ on $\partial \Omega$. Then

$$\begin{aligned} \Vert {\widetilde{\partial }} T^K {\widetilde{\partial }} q\Vert _{L^2(\Omega )}\lesssim & {} c_0{\sum }_{S\in {\mathcal {S}}}\Vert {\widetilde{\partial }} {S} T^{K} {\widetilde{x}}\Vert _{L^2(\Omega )} +c_0\Vert {\widetilde{{\text {div}}}} \big ( T^{K} {\widetilde{\partial }} q\big )\Vert _{L^2(\Omega )}\nonumber \\{} & {} +c_K{\sum }_{|L|\le |K|-1}\Vert {\widetilde{{\text {div}}}} \big ( T^{L} {\widetilde{\partial }} q\big )\Vert _{L^2(\Omega )}\nonumber \\{} & {} +c_K{\sum }_{|K'|\le |K|}\Vert {\widetilde{\partial }} T^{K'} {\widetilde{x}}\Vert _{L^2(\Omega )} , \end{aligned}$$

(B.3)

and

(B.4)

where $c_K$ depends on ${\widetilde{\partial }} T^{N}{\widetilde{x}}$ and ${\widetilde{\partial }} T^{N}{\widetilde{\partial }}q$, for $|N|\le |K|/2+3$.

Lemma B.7 is a consequence of the following two lemmas, Lemma B.1 applied $\alpha =T^K {\widetilde{\partial }} q$, and induction.

Lemma B.8

Suppose that $q=0$ on $\partial \Omega$. Then

$$\begin{aligned} \Vert S T^K {\widetilde{\partial }} q\Vert _{L^2(\Omega )}\lesssim & {} c_0{\sum }_{S\in {\mathcal {S}}}\Vert {\widetilde{\partial }} {S} T^{K} {\widetilde{x}}\Vert _{L^2(\Omega )} +c_0\Vert {\widetilde{{\text {div}}}} \big ( T^{K} {\widetilde{\partial }} q\big )\Vert _{L^2(\Omega )}\\{} & {} +c_K{\sum }_{|L|\le |K|-1}\Vert {\widetilde{\partial }} T^{L} {\widetilde{\partial }} q\Vert _{L^2(\Omega )} +c_K{\sum }_{|K'|\le |K|}\Vert {\widetilde{\partial }} T^{K'} {\widetilde{x}}\Vert _{L^2(\Omega )} , \end{aligned}$$

and

where $c_K$ depends on ${\widetilde{\partial }} T^{N}{\widetilde{x}}$ and ${\widetilde{\partial }} T^{N}{\widetilde{\partial }}q$, for $|N|\le |K|/2+3$.

Lemma B.9

Let $A_{ij}^J={\widetilde{\partial }}_i T^J {\widetilde{\partial }}_j q-{\widetilde{\partial }}_j T^J {\widetilde{\partial }}_i q$. We have

(B.5)

where $c_K$ stands for a constant that depends on ${\widetilde{\partial }} T^N {\widetilde{x}}$ and ${\widetilde{\partial }} T^{N}{\widetilde{\partial }}q$, for $|N|\le |K|/2$.

Moreover let $A_{ij}^{J,\nicefrac {1}{2}} ={\widetilde{\partial }}_i \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}T^J {\widetilde{\partial }}_j q-{\widetilde{\partial }}_j \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}T^J {\widetilde{\partial }}_i q$. Then

(B.6)

1.1.2 B.0.3. The proofs of the basic elliptic estimates

Proof of Lemma B.4

Integrating by parts:

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

(B.7)

We insert the identity:

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

into the first term in (B.7) and integrate by parts again:

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

Note that by the antisymmetry of curl:

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

so (B.7) becomes:

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

Here:

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

$\square$

Proof of Lemma B.5

We have the following identity

$$\begin{aligned} \partial _i\big ( \alpha ^i \alpha _j {\mathcal {N}}^j \big ) - \partial _j\big ( \alpha ^i \alpha _i\, {\mathcal {N}}^j \big )/2{} & {} ={\text {div}}(\alpha )\, \alpha _{j}{\mathcal {N}}^j+ {\text {curl}}\,\alpha _{ij}\, \alpha ^i {\mathcal {N}}^j+\alpha ^i \alpha ^j \partial _i {\mathcal {N}}_j \\{} & {} \quad -|\alpha |^2 \partial _j {\mathcal {N}}^j /2. \end{aligned}$$

Integrating this over the domain gives the lemma. $\square$

Proof of Lemma B.8

Integrating by parts we get

$$\begin{aligned} \int _\Omega S S T^K\! q\, \, {\widetilde{{\text {div}}}} \big ( T^K {\widetilde{\partial }} q\big ) d{\widetilde{x}}= & {} - \int _\Omega {\widetilde{\partial }}_i S S T^K\! q\, \, T^K {\widetilde{\partial }}^i q d{\widetilde{x}}\\= & {} -\int _\Omega S{\widetilde{\partial }}_i S T^K \!q\, \, T^K {\widetilde{\partial }}^i q\, d{\widetilde{x}}\\{} & {} +\int _\Omega {\widetilde{\partial }}_i S{\widetilde{x}}^k\, \, {\widetilde{\partial }}_k S T^K \! q\, \, T^K {\widetilde{\partial }}^i q\, d{\widetilde{x}}\\= & {} \int _\Omega {\widetilde{\partial }}_i S T^K\!q\, \,(S+{\widetilde{{\text {div}}}} S) T^K {\widetilde{\partial }}^i q\, d{\widetilde{x}} \\{} & {} +\int _\Omega {\widetilde{\partial }}_i S{\widetilde{x}}^k\, \, {\widetilde{\partial }}_k S T^K \! q\, \, T^K {\widetilde{\partial }}^i q\, d{\widetilde{x}} . \end{aligned}$$

The proof of the first inequality follows from this and

$$\begin{aligned} {\widetilde{\partial }}_i T^J \! q -\! T^J{\widetilde{\partial }}_i q= & {} R_i^J\!,\quad \text {where}\quad R_i^J\!={\widetilde{\partial }}_i T^J{\widetilde{x}}^k\,\,{\widetilde{\partial }}_k q\\{} & {} +\!\!\!\!\!\!\sum _{J_1+\dots +J_k=J,\,|J_i|<|J|}\!\!\!\!\! r_{\!J_1\dots J_k}^J\!{\widetilde{\partial }}_i T^{J_1} {\widetilde{x}}\cdots \! {\widetilde{\partial }}T^{J_{k-1}} {\widetilde{x}}\cdot T^{J_k} {{\widetilde{\partial }}}^{i} q. \end{aligned}$$

To prove the second inequality we integrate by parts again

$$\begin{aligned}{} & {} \int _\Omega S \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}S T^K q\, \, {\widetilde{{\text {div}}}} \big (\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}T^K {\widetilde{\partial }} q\big ) d{\widetilde{x}}\\{} & {} \quad =- \int _\Omega {\widetilde{\partial }}_i\big ( S \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}S T^K q\big )\, \,\big (\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}T^K {\widetilde{\partial }}^i q\big ) d{\widetilde{x}}\\{} & {} \quad =-\int _\Omega S{\widetilde{\partial }}_i\big (\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}S T^K q\big )\, \,\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}T^K {\widetilde{\partial }}^i q\, d{\widetilde{x}}\\{} & {} \qquad +\int _\Omega {\widetilde{\partial }}_i S{\widetilde{x}}^k\, \, {\widetilde{\partial }}_k\big (\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}S T^K q\big )\, \,\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}T^K {\widetilde{\partial }}^i q\, d{\widetilde{x}}\\{} & {} \quad =\int _\Omega {\widetilde{\partial }}_i\big ( \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}S T^K q\big )\, \,(S+{\widetilde{{\text {div}}}} S)\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}T^K {\widetilde{\partial }}^i q\, d{\widetilde{x}} \\{} & {} \qquad +\int _\Omega {\widetilde{\partial }}_i S{\widetilde{x}}^k\, \, {\widetilde{\partial }}_k\big (\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}S T^K q\big )\, \,\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}T^K {\widetilde{\partial }}^i q\, d{\widetilde{x}} . \end{aligned}$$

Here by Lemma A.10

Using Lemma A.9 we get

$$\begin{aligned} \Vert \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}R^J\Vert _{L^2(\Omega )}\lesssim & {} c_J \!\!\!\!\sum _{|K|\le |J|-1} \Vert \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}T^K {\widetilde{\partial }} q \Vert _{L^2(\Omega )}\\{} & {} +c_J \!\!\!\! \sum _{|N|\le |J|/2+3}\!\!\!\!\!\!\Vert T^N {\widetilde{\partial }} q\Vert _{L^\infty (\Omega )} \!\! \!\!\sum _{|J'|\le |J|}\!\!\!\!\Vert {\widetilde{\partial }}\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}T^{J'} {\widetilde{x}}\Vert _{L^2(\Omega )}, \end{aligned}$$

where $c_J$ depends on ${\widetilde{\partial }} T^{N}{\widetilde{x}}$ for $|N|\le |J|/2+3$. The lemma follows from these estimates and induction. $\square$

Proof of Lemma B.9

Recall that for some constants $a^K_{K_1\dots K_k}$

$$\begin{aligned} A^K_{ij}= & {} {\widetilde{\partial }}_i T^K{\widetilde{x}}^k\,\,{\widetilde{\partial }}_k {\widetilde{\partial }}_j q -{\widetilde{\partial }}_j T^K{\widetilde{x}}^k\,\,{\widetilde{\partial }}_k {\widetilde{\partial }}_i q \\{} & {} +{\sum }_{K_1+\dots +K_k=K,\, |K_i|<|K|}\,\, a^K_{K_1\dots K_k} {\widetilde{\partial }}_i T^{K_1} {\widetilde{x}}\cdots \! {\widetilde{\partial }}T^{K_{k-1}} {\widetilde{x}}\cdot \!{\widetilde{\partial }} T^{K_k} {{\widetilde{\partial }}}^{i} q, \end{aligned}$$

from which (B.5) follows. By Lemma A.10

$$\begin{aligned} \Vert A^{K,\nicefrac {1}{2}}\Vert _{L^2(\Omega )}\lesssim \Vert \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}A^K\Vert _{L^2(\Omega )} +c_0\Vert {\widetilde{\partial }} T^K {\widetilde{\partial }} q\Vert _{L^2(\Omega )}, \end{aligned}$$

and by Lemma A.9 and Lemma A.10

which proves (B.6). $\square$

Proof of Lemma B.6

By (B.1) we have for $k=0,1$:

$$\begin{aligned} |{\widetilde{\partial }}\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {k}{2}} T^K {\widetilde{\partial }} h| \lesssim |_{\,}{\widetilde{{\text {div}}}}\,\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {k}{2}} T^K {\widetilde{\partial }} h|+|_{\,}{\widetilde{{\text {curl}}}}\,\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {k}{2}} T^K {\widetilde{\partial }} h| + {\sum }_{S\in {{\mathcal {S}}}}|S\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {k}{2}} T^K {\widetilde{\partial }} h|. \end{aligned}$$

(B.3) follows from this for $k=0$, Lemma B.9, Lemma B.8 and induction to deal with the lower order term. The proof of (B.4) follows in the same way apart from that we also have to use (B.3). $\square$

Appendix C. Basic elliptic estimates with respect to the Lorentz metric g

In this section we prove some generalizations of the estimates from Section B.

For a one-form $\beta = \beta _\mu d{\widetilde{x}}^\mu$ we write

$$\begin{aligned} {\widetilde{{\text {div}}}} \beta = {\widetilde{\nabla }}^\mu \beta _\mu , \qquad {\widetilde{{\text {curl}}}}\beta _{\mu \nu } = {\widetilde{\nabla }}_\mu \beta _\nu - {\widetilde{\nabla }}_\nu \beta _\mu = {\widetilde{\partial }}_\mu \beta _\nu - {\widetilde{\partial }}_\nu \beta _\mu , \end{aligned}$$

where in the last step we used the symmetry of the Christoffel symbols. Let denote the future-directed timelike vector defining the time axis of the background spacetime (g, M),

We will work in terms of the following Riemannian metric on the spacetime M,

For one-forms $\alpha$ and two-forms $\omega$ we will use the pointwise norms

$$\begin{aligned} |\alpha |^2= & {} H^{\mu \nu } \alpha _\mu \alpha _\nu , \qquad \\ |\omega |^2= & {} H^{\mu \nu }H^{\alpha \beta } \omega _{\mu \alpha }\omega _{\nu \beta }, \end{aligned}$$

We have the following pointwise estimate.

Lemma C.1

There is a constant $c_0 = c_0(|\partial {\widetilde{x}}|)$ so that for any one-form $\beta$ on ${\mathcal {D}}$ we have

$$\begin{aligned} |{\widetilde{\partial }}\beta | \le c_0\big ( |{\widetilde{{\text {div}}}} \beta | + |{\widetilde{{\text {curl}}}} \beta | + |{\mathcal {S}}\beta | + |\beta |\big ). \end{aligned}$$

(C.1)

Recall that ${\mathcal {S}}$ runs over the family of spacetime vector fields which are tangent to $\partial \Omega$.

We will also need some elliptic estimates on the surfaces $\Omega _{s'} =\Omega \times \{s = s'\}$ of constant $s'$. For this we work in terms of the Riemannian metric G defined in (3.16) which we recall here.

which satisfies

$$\begin{aligned} G(\xi ,\xi ) \ge c {\overline{g}}(\xi ,\xi ), \end{aligned}$$

(C.2)

for a constant c (see (3.17)), for any vector $\xi \in T\Omega _{s'}$.

For one-forms X and two-tensors $\omega$ we write

$$\begin{aligned} {\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert X \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }_{L^2(\Omega )}^2 = \int _{\Omega } G^{\mu \nu } X_\mu X_\mu \, \kappa _{G} dy, \quad {\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \omega \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }_{L^2(\Omega )}^2 = \int _{\Omega } G^{\alpha \beta } G^{\mu \nu } \omega _{\alpha \mu }\omega _{\beta \nu } \,\kappa _{G} dy. \qquad \end{aligned}$$

(C.3)

Here, $\kappa _{G} = \det G^{1/2}$. Then ${\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert X \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }_{L^2(\Omega )}$ is positive definite when restricted to one-forms X which are cotangent to $\Omega$ at fixed s.

We will also work in terms of covariant differentiation ${\overline{\nabla }}$ with respect to the metric G which satisfies ${\overline{\nabla }}G =0$. If X is a one-form then it is given by

$$\begin{aligned} {\overline{\nabla }}_\mu X_\nu = G^{\mu '}_{\mu } G_{\nu }^{\nu '} \nabla _{\mu '}X_{\nu '}. \end{aligned}$$

Here, $G^{\mu }_{\nu }$ denotes orthogonal projection to the tangent space of $\Omega$ with respect to the metric G,

$$\begin{aligned} G^{\mu }_{\nu } = g_{\nu \nu '} G^{\mu \nu '}. \end{aligned}$$

We also write

$$\begin{aligned} {\text {div}}_{G} X = G^{\mu \nu } {\overline{\nabla }}_\mu X_\nu . \end{aligned}$$

(C.4)

and we have

Lemma C.2

There is a constant $C_{\!1}$ depending on ${\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \partial {\widetilde{x}} \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }_{L^\infty (\Omega )}$ as well as $c_L$ from (C.2) so that with notation as in (C.3), if X is a one-form on $\Omega$

$$\begin{aligned} \! {\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert {\widetilde{\partial }}X \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }_{L^{2{}_{\!}}(\Omega )}^2\!\le & {} C_{{}_{\!}1\!}\Big (\! {\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert {\text {div}}_{G} X \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }_{L^2(\Omega )}^2\! +{}_{\!} {\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert {\widetilde{{\text {curl}}}}X \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }_{L^2(\Omega )}^2\! \nonumber \\{} & {} + \! \!\int _{\partial \Omega } (G^{\mu \nu } n_{{}_{\!}\mu } \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}\! X_\nu ) \cdot (G^{\alpha \beta } n_{{}_{\!}\alpha } \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}\! X_\beta ) dS\nonumber \\{} & {} + {\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert X \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }_{L^2(\partial \Omega )}^2\! + {\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert X \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }_{L^2(\Omega )}^2\!\Big ). \end{aligned}$$

(C.5)

Here $\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}$ is a half angular derivative defined locally in coordinates in (A.9), and the inner product is the sum over coordinate charts $\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}\xi ^a \cdot \langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}\xi ^b=\sum _{\mu } \big (\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu \xi ^a\big ) \big (\langle \partial _{\theta {}_{\!}} \rangle ^{\!\nicefrac {1}{2}}_\mu \xi ^b\big )$ as in (A.10).

The next result is similar to Proposition B.6 and follows in almost exactly the same way.

Proposition C.3

Suppose that $q=0$ on $\partial \Omega$. Then with notation as in (C.3),

and

where $C_K\!\!$ depends on $\Vert {\widetilde{\partial }} T^{L}{\widetilde{x}}\Vert _{L^\infty (\Omega )}$, $\Vert {\widetilde{\partial }} T^{L}{\widetilde{\partial }}q\Vert _{L^\infty (\Omega )}$,$\Vert {\widetilde{\partial }} T^L {\widetilde{H}}\Vert _{L^\infty }\!$ for $|L|\!\!\le \!\! |K|/2\!+\!3$, and $c_L$ from (C.2).

1.1 Proofs of the basic elliptic estimates used in the relativistic case

Proof of Lemma C.1

This is similar to the proof of the pointwise lemma from [16]. First, we note that the metric H is equivalent to the metric $h^{\mu \nu } = {\widetilde{g}}^{\mu \nu } + 2{\widetilde{u}}^\mu {\widetilde{u}}^\nu$ and so it is enough to prove (C.1) with all pointwise norms replaced by the norms with respect to h. It is more convenient to state the result in terms of H since that does not depend on the fluid variables but for the proof it is better to work with h since it is more clear how the material derivatives enter. Let ${\widetilde{{\mathcal {N}}}}$ denote the unit normal with respect to h to $\partial \Omega$ (at constant s) and extend it to a tubular neighborhood of the boundary. Since the right-hand side of (C.1) controls the material derivative $D_s \beta$ and since $D_s = {\widetilde{V}}^\mu {\widetilde{\partial }}_\mu$ is parallel to ${\widetilde{u}}^\mu {\widetilde{\partial }}_\mu$, it is enough to prove that if $\omega = \omega _{\alpha \beta }d{\widetilde{x}}^\alpha d{\widetilde{x}}^\beta$ is a symmetric two-tensor satisfying ${\widetilde{g}}^{\alpha \beta }\omega _{\alpha \beta } = 0$ and ${\widetilde{u}}^\alpha {\widetilde{u}}^\beta \omega _{\alpha \beta } = 0$ then

$$\begin{aligned} {\widetilde{g}}^{\alpha \beta }{\widetilde{g}}^{\mu \nu } \omega _{\alpha \mu }\omega _{\beta \nu } \le C q^{\alpha \beta } {\widetilde{g}}^{\mu \nu }\omega _{\alpha \mu }\omega _{\beta \nu }, \end{aligned}$$

where $q^{\alpha \beta } = h^{\alpha \beta } - {\widetilde{{\mathcal {N}}}}^\alpha {\widetilde{{\mathcal {N}}}}^\beta$ is the projection onto the orthogonal complement to ${\widetilde{{\mathcal {N}}}}$.

Writing $h^{\alpha \beta } = q^{\alpha \beta }+{\widetilde{{\mathcal {N}}}}^a {\widetilde{{\mathcal {N}}}}^b$ and using the symmetry of $\omega$ as well as the fact that the component of h along u annihilates $\omega$, we have

$$\begin{aligned} {\widetilde{g}}^{\alpha \beta }{\widetilde{g}}^{\mu \nu }\omega _{\alpha \mu } \omega _{\beta \nu }= & {} q^{\alpha \beta } q^{\mu \nu }\omega _{\alpha \mu }\omega _{\beta \nu } + {\widetilde{N}}^\alpha {\widetilde{N}}^\beta {\widetilde{N}}^\mu {\widetilde{N}}^\nu \omega _{\alpha \mu }\omega _{\beta \nu }\nonumber \\{} & {} + 2 q^{\alpha \beta } {\widetilde{N}}^\mu {\widetilde{N}}^\nu \omega _{\alpha \mu }\omega _{\beta \nu } \end{aligned}$$

(C.6)

If $\omega$ additionally satisfies ${\widetilde{g}}^{\alpha \mu } \omega _{\alpha \mu } = 0$ then the second term on the first line is

$$\begin{aligned} {\widetilde{N}}^\alpha \! {\widetilde{N}}^\beta \! {\widetilde{N}}^\mu \! {\widetilde{N}}^\nu \! \omega _{\alpha \mu }\omega _{\beta \nu }= & {} \!\big ( {\widetilde{N}}^\alpha \!{\widetilde{N}}^\mu \omega _{\alpha \mu }\big )^2\!\! = \big ( {\widetilde{g}}^{\alpha \mu } \omega _{\alpha \mu } - q^{\alpha \mu } \omega _{\alpha \mu }\big )^2\!\! \\= & {} \! \big ( {\widetilde{g}}^{\alpha \mu } \omega _{\alpha \mu }\big )^2\! + \big ( q^{\alpha \mu } \omega _{\alpha \mu }\big )^2\! - 2 {\widetilde{g}}^{\alpha \mu } \omega _{\alpha \mu }q^{\beta \nu } \omega _{\beta \nu }. \end{aligned}$$

Inserting this identity into (C.6) we have

$$\begin{aligned} {\widetilde{g}}^{\alpha \beta }{\widetilde{g}}^{\mu \nu }\omega _{\alpha \mu } \omega _{\beta \nu }= & {} q^{\alpha \beta } q^{\mu \nu }\omega _{\alpha \mu }\omega _{\beta \nu } +2 q^{\alpha \beta } {\widetilde{N}}^\mu {\widetilde{N}}^\nu \omega _{\alpha \mu }\omega _{\beta \nu }\\{} & {} + \big ( {\widetilde{g}}^{\alpha \mu } \omega _{\alpha \mu }\big )^2 + \big ( q^{\alpha \mu } \omega _{\alpha \mu }\big )^2 - 2 {\widetilde{g}}^{\alpha \mu } \omega _{\alpha \mu }q^{\beta \nu } \omega _{\beta \nu }. \end{aligned}$$

Using the symmetry of $\omega$ we have $(q^{\alpha \mu }\omega _{\alpha \mu })^2 \le C q^{\alpha \beta } q^{\mu \nu } \omega _{\alpha \mu }\omega _{\beta \nu }$ and this gives the result. $\square$

Proof of Lemma C.2

In the same way that Lemma B.4 implied (B.2), Lemma C.2 is a consequence of the following identity after noting that the boundary term only involves derivatives which are tangent to $\partial \Omega$. We recall the definition of the norms ${\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \beta \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }_{L^2}$ from (3.12).

Lemma C.4

Let $n_\mu$ denote the spacelike unit conormal to $\partial \Omega$ normalized with respect to the metric $G$, defined in (3.16). If X is a one-form, then

Here, denotes covariant differentiation tangent to $\partial \Omega$ with s held constant, given by

We are also writing $R_G$ for the Ricci curvature tensor of G and

$$\begin{aligned} {\text {curl}}X_{\mu \nu } = \nabla _\mu X_\nu - \nabla _\nu X_\mu ={\widetilde{\partial }}_\mu X_\nu - {\widetilde{\partial }}_\nu X_\mu . \end{aligned}$$

Lemma C.4 is proven in essentially the same way that we proved Lemma B.4. We start by recording the divergence theorem in terms of ${\text {div}}_G$,

$$\begin{aligned} \int _{\Omega } {\text {div}}_G X \, \kappa _G dy = \int _{\partial \Omega } G^{\mu \nu } n_\mu X_\nu \, dS_G, \end{aligned}$$

(C.7)

where $\kappa _G dy$ is the Riemannian volume element with respect to G and $dS_G$ denotes the corresponding surface measure, and $n_\mu$ is the unit conormal to $\partial \Omega$ normalized with respect to G.

Using (C.7) along with the fact that ${\overline{\nabla }}G = 0$,

$$\begin{aligned} \int _{\Omega } G^{\mu \nu } G^{\alpha \beta } {\overline{\nabla }}_{\!\mu } X_\alpha {\overline{\nabla }}_{\!\nu } X_\beta \, \kappa _G dy= & {} -\!\!\int _{\Omega }\!\! \big (G^{\mu \nu }{\overline{\nabla }}_{\!\nu }{\overline{\nabla }}_{\!\mu } X_\alpha \big ) G^{\alpha \beta } X_\beta \,\kappa _G dy \nonumber \\{} & {} +\!\! \int _{\partial \Omega }\!\!\! G^{\mu \nu }G^{\alpha \beta }n_\nu X_\beta {\overline{\nabla }}_\mu X_\alpha \,dS_G. \end{aligned}$$

(C.8)

We have the identity

$$\begin{aligned} G^{\mu \nu }{\overline{\nabla }}_\nu {\overline{\nabla }}_\mu X_\alpha= & {} G^{\mu \nu } {\overline{\nabla }}_\alpha {\overline{\nabla }}_\nu X_\mu + G^{\mu \nu } {\overline{\nabla }}_\mu ({\overline{\nabla }}_\nu X_\alpha - {\overline{\nabla }}_\alpha X_\nu ) + G^{\mu \nu }R_{G \alpha \mu \nu }^\beta X_{\beta } \nonumber \\= & {} {\overline{\nabla }}_\alpha ({\text {div}}_G\, X) + G^{\mu \nu }{\overline{\nabla }}_\mu {\overline{{\text {curl}}}} X_{\nu \alpha } + G^{\mu \nu }R_{G \alpha \mu \nu }^\beta X_{\beta } , \end{aligned}$$

(C.9)

where $R_G$ denotes the curvature tensor of G,

$$\begin{aligned} R_{G\alpha \mu \nu }^\beta X_\beta = [{\overline{\nabla }}_\alpha {\overline{\nabla }}_\mu - {\overline{\nabla }}_\mu {\overline{\nabla }}_\alpha ] X_\nu . \end{aligned}$$

Inserting (C.9) into the first term on the right of (C.8) and integrate by parts again:

$$\begin{aligned} \int _{\Omega }\int _{\Omega } G^{\alpha \beta } X_\beta \big (G^{\mu \nu }{\overline{\nabla }}_\mu {\overline{\nabla }}_\nu X_\alpha \big )= & {} -\int _{\Omega } ({\text {div}}_{G}\, X)^2 + G^{\mu \nu } G^{\alpha \beta } {\overline{\nabla }}_\mu X_\beta {\overline{{\text {curl}}}} X_{\nu \alpha } \\{} & {} +\int _{\partial \Omega } G^{\alpha \beta } n_\alpha X_\beta {\text {div}}_G\, X +G^{\mu \nu } G^{\alpha \beta } n_\mu {\overline{{\text {curl}}}} X_{\nu \alpha } X_\beta \, dS_{G}\\{} & {} -\int _{\Omega }G^{\mu \nu }G^{\alpha \beta } R_{G \alpha \mu \nu }^\gamma X_{\gamma } X_\beta \end{aligned}$$

By the antisymmetry of curl, $G^{\mu \nu }G^{\alpha \beta } {\overline{\nabla }}_{\!\mu } X_\beta {\overline{{\text {curl}}}} X_{\nu \alpha } = \frac{1}{2} G^{\mu \nu }G^{\alpha \beta } {\overline{{\text {curl}}}} X_{\mu \beta } {\overline{{\text {curl}}}} X_{\nu \alpha },$ so (C.8) becomes

$$\begin{aligned} \int _{\Omega } G^{\mu \nu } G^{\alpha \beta } {\overline{\nabla }}_\mu X_\alpha {\overline{\nabla }}_\nu X_\beta= & {} \int _{\Omega }({\text {div}}_{G}\, X)^2 + \frac{1}{2} G^{\mu \nu } G^{\alpha \beta } {\overline{{\text {curl}}}} X_{\mu \beta } {\overline{{\text {curl}}}} X_{\nu \alpha }\\{} & {} + \int _{{\widetilde{\partial }}\Omega } G^{\mu \nu }G^{\alpha \beta } X_\beta n_\mu {\overline{\nabla }}_\nu X_\alpha - G^{\alpha \beta } n_\alpha X_\beta {\text {div}}_{G}\, X \\{} & {} - G^{\mu \nu }G^{\alpha \beta } n_\mu X_\beta {\overline{{\text {curl}}}} X_{\nu \alpha } -\int _{\Omega }G^{\mu \nu }G^{\alpha \beta } R_{G \alpha \mu \nu }^\gamma X_{\gamma } X_\beta . \end{aligned}$$

We now use that $G^{\mu \nu }n_\mu n_\nu = 1$ to write

Using this expression, the boundary term is the integral of

Noting that the terms on the second line cancel, we get the result. $\square$

Appendix D. The divergence theorem

The identity (3.20) is nothing but the usual divergence theorem, see e.g. [4]. If ${\widetilde{D}}$ denotes intrinsic covariant differentiation on $\Lambda$,

$$\begin{aligned} {\text {div}}_{\Lambda }\, T = {\widetilde{D}}_\mu T^\mu . \end{aligned}$$

and the divergence theorem on $\Lambda$ says

$$\begin{aligned} \int _{\Lambda _{\Sigma _0}^{\Sigma _1}} {\text {div}}_{\Lambda }\, T\, dS^{\Lambda } = \int _{\Lambda _{\Sigma _1}} {\widetilde{g}}(n^{\Sigma _1}, T) \,dS^{\Lambda _{\Sigma _1}} + \int _{\Lambda _{\Sigma _0}} {\widetilde{g}}(n^{\Sigma _0}, T) \,dS^{\Lambda _{\Sigma _0}}. \end{aligned}$$

(D.1)

where $n^{\Sigma _i}$ denotes the future-directed normal vector field to $\Sigma _i$ defined relative to ${\widetilde{g}}$ and $\Lambda _{\Sigma _i}= \Lambda \cap \Sigma _i$. If ${\widetilde{V}}^\mu$ is tangent to $\Lambda$ then with $D_s = {\widetilde{V}}^\mu {\widetilde{\partial }}_\mu$,

$$\begin{aligned} D_s\phi = {\widetilde{V}}^\mu {\widetilde{\partial }}_\mu \phi = {\widetilde{V}}^\mu {\widetilde{D}}_\mu \phi = {\text {div}}_{\Lambda }\,({\widetilde{V}} \phi ) - \phi {\text {div}}_{\Lambda }\, {\widetilde{V}}, \end{aligned}$$

and integrating this expression and using (D.1) gives (3.21).

Appendix E. Existence for the linear and smoothed problem

In this section we give a sketch of the proof of existence for the linear problems we use in our iteration scheme. Since this is a linear problem with tangentially smoothed coefficients, existence on a time interval depending on the smoothing parameter is nearly an immediate consequence of the a priori estimates we proved in the earlier sections. We first discuss the Newtonian case.

1.1 Existence for the linear and smoothed Newtonian problem

Fix a tangentially smooth vector field ${\widetilde{V}}$ and define ${\widetilde{x}}$ by

$$\begin{aligned} \frac{d{\widetilde{x}}(t,y)}{dt} ={\widetilde{V}}(t,y), \qquad {\widetilde{x}}(0,y) = y. \end{aligned}$$

The linear problem we consider is

$$\begin{aligned} D_t V_i + {\widetilde{\partial }}_i h[V] = 0, \qquad \text { in } [0,T_1]\times \Omega , \qquad V|_{t = 0} = V_0, \end{aligned}$$

(E.1)

with ${\widetilde{\partial }}_i = \frac{\partial }{\partial {\widetilde{x}}_i} = \frac{\partial y^a}{\partial {\widetilde{x}}^i} \frac{\partial }{\partial y^a}$, and where $h = h[V]$ is determined by solving the wave equation

$$\begin{aligned} e_1D_t^2 h - {\widetilde{\Delta }} h = ({\widetilde{\partial }}_i {\widetilde{V}}^j)({\widetilde{\partial }}_j V^i), \quad h|_{\partial \Omega } = 0, \quad h|_{t = 0} = h_0, \quad D_t h|_{t = 0} = h_1. \qquad \end{aligned}$$

(E.2)

Here ${\widetilde{\Delta }} = \delta ^{ij}{\widetilde{\partial }}_i {\widetilde{\partial }}_j$.

To solve (E.1) we are going to show that it is an ODE in a certain function space (for $\varepsilon \!>\! 0)$, and existence then follows from a standard Picard iteration. Fix $r\!\ge \! 10$ and for $T_0\! >\! 0$, define the norms

$$\begin{aligned} \Vert u\Vert _{X_{T_0}} ={\sup }_{0 \le t \le T_0} \Vert u(t)\Vert _{r+1}, \end{aligned}$$

where

$$\begin{aligned} \Vert u(t)\Vert _{r+1} = {\sum }_{k+\ell \le r}\Vert D_t D_t^k u(t)\Vert _{H^{\ell }(\Omega )} + {\sum }_{k+\ell \le r}\Vert D_t^k u(t)\Vert _{H^{\ell }(\Omega )}. \end{aligned}$$

(E.3)

The reason we work with norms that control one additional time derivative will be explained in section E.1.1. In that section we show that the map $V\mapsto h[V]$ is well-defined if the compatibility conditions hold and $\Vert {\widetilde{V}}\Vert _{X_{T_1}} \!+\! \Vert {\mathcal {S}}{\widetilde{x}}\Vert _{X_{T_1}} < \infty$, where ${\mathcal {S}}{\widetilde{x}}$ is defined in (A.3) and involves tangential derivatives of ${\widetilde{x}}$. With

$$\begin{aligned} H[V](t,y) = -\int _0^t {\widetilde{\partial }}h[V](t',y)\, dt'. \end{aligned}$$

in Section E.1.1, we show that H is bounded and Lipschitz on $X_{T_1}$,

$$\begin{aligned} \Vert H[V]\Vert _{X_{T_1}}&\le C( \Vert {\widetilde{V}}\Vert _{X_{T_1}}, \Vert {\mathcal {S}}{\widetilde{x}}\Vert _{X_{T_1}}) \left( \Vert {\overline{V}}\Vert _{X_{T_1}} + T_1 \Vert V\Vert _{X_{T_1}}\right) , \end{aligned}$$

(E.4)

$$\begin{aligned} \Vert H[V_1] - H[V_2]\Vert _{X_{T_1}}&\le T_1C( \Vert {\widetilde{V}}\Vert _{X_{T_1}}, \Vert {\mathcal {S}}{\widetilde{x}}\Vert _{X_{T_1}}, \Vert V_1\Vert _{X_{T_1}}, \Vert V_2\Vert _{X_{T_1}}) \Vert V_1 - V_2\Vert _{X_{T_1}}. \end{aligned}$$

(E.5)

In (E.4), ${\overline{V}}$ is a power series in time which solves the equation at $t = 0$ to order r (see (E.8)) and is determined from the initial data $V_0, h_0$ and satisfies $\Vert {\overline{V}}\Vert _{X_{T_1}} \lesssim \Vert V_0\Vert _{H^r(\Omega )} + \Vert {\widetilde{\partial }}h_0\Vert _{H^{r-1}(\Omega )}.$

Assuming these bounds hold, existence follows from a straightforward Picard iteration.

Proposition E.1

(Existence for the linear and smoothed problem) Let $r \ge 10$ and suppose that the initial data $(V_0, h_0)$ satisfies the compatibility conditions (E.10) to order r. Let ${\widetilde{V}} \in X_{T_1}$ for some $T_1 > 0$. Then there is a time $T \le T_1$ so that the linear smoothed problem (E.1) has a unique solution $V \in X_{T}$ and if ${\overline{V}}$ denotes a formal power series solution at $t = 0$ defined as in (E.8), V satisfies the bound

$$\begin{aligned} \Vert V\Vert _{X_{T}} \le C\big (\Vert {\widetilde{V}}\Vert _{X_{T_1}} \Vert {\mathcal {S}}{\widetilde{x}}\Vert _{X_{T_1}}\big ) \Vert {\overline{V}}\Vert _{X_{T}}, \end{aligned}$$

(E.6)

and the enthalpy satisfies

$$\begin{aligned}{} & {} \sup _{0 \le t \le T} \sum _{k +\ell \le r-1}\!\!\!\! \Vert D_t^k {\widetilde{\partial }}h(t)\Vert _{H^\ell (\Omega )} +\Vert D_t^k D_t h(t)\Vert _{H^\ell (\Omega )} \nonumber \\{} & {} \quad \le C \big (\!\!\!\!\sum _{k +\ell \le r-1}\!\!\!\! \Vert D_t^k {\widetilde{\partial }}h(0)\Vert _{H^\ell (\Omega )} +\Vert D_t^k D_t h(0)\Vert _{H^\ell (\Omega )}\big ), \end{aligned}$$

(E.7)

with $C \!= \!C\big (\Vert {\widetilde{V}}\Vert _{X_{T_1}}, \Vert {\mathcal {S}}{\widetilde{x}}\Vert _{X_{T_1}}\!\big )$. Moreover, the compatibility conditions hold at time $t \!=\! T\!$ to order r.

Proof

We are going to solve (E.1) by iteration and so we need to ensure that the map $V\mapsto h[V]$ is well-defined at each step. In particular we need to ensure that if V satisfies the compatibility conditions from the upcoming section then so does the resulting W. We therefore work in the space

$$\begin{aligned} X_{T_1,c} = \{V : \Vert V\Vert _{X_{T_1}} < \infty , D_t^k V|_{t = 0} = V_k, k = 0,..., r+1\}, \end{aligned}$$

where the $V_k$ are given by (E.9). We claim that if $V \in X_{T, c}$ and W satisfies $D_t W = -{\widetilde{\partial }}h[V]$ then $W \in X_{T_1, c}$ as well. First, by the results of the upcoming section E.1.1 given $V \in X_{T_1, c}$, ${\widetilde{\partial }}h[V]$ is well-defined and by (E.4) the resulting W with $D_t W = -{\widetilde{\partial }}h$ satisfies the bound (E.4). It remains to check the time derivatives at $t = 0$. For these we compute

$$\begin{aligned} D_t^{k}V'|_{t = 0} = D_t^{k-1} {\widetilde{\partial }}h[V]|_{t = 0} = V_k, \end{aligned}$$

which is just the definition of the $V_k$. Using the bounds (E.4)-(E.5), the existence result and the bounds follow by a standard iteration argument. The fact that the compatibility conditions hold at later times as well follows directly from the construction of the enthalpy, see section E.4. $\square$

It remains to prove that under the hypotheses of the above Proposition, the map $V \mapsto h[V]$ is well-defined and that (E.4)-(E.5) hold. This is done in the next section.

1.1.1 The compatibility conditions and existence for the wave equation for the enthalpy

Because of the continuity equation and that $h = 0$ on $\partial \Omega$, the initial data $V_0$ must satisfy ${\widetilde{{\text {div}}}} V_0 = 0$ on $\partial \Omega$. Taking more time derivatives we see that we must also have $D_t^k ({\widetilde{{\text {div}}}} V)|_{t = 0} = 0$ on the boundary which places additional restrictions on the initial data that we now write out explicitly.

Fix a diffeomorphism $x_0:\Omega \rightarrow \Omega$. Let ${\overline{V}} = \sum _{k\ge 0} V_k{t^k}/{k!}$, ${\overline{h}} = \sum _{k \ge 0} h_k{t^k}/{k!}$, and ${\overline{x}} = x_0 + t{\overline{V}}$ be a formal power series solution to (4.2)-(4.4) at $t= 0$,

$$\begin{aligned} D_t^k\big (D_t {\overline{V}} + {\widetilde{\partial }} {\overline{h}}\big )|_{t = 0} = 0, \qquad D_t^k \big ( e_1 D_t {\overline{h}} + {\widetilde{{\text {div}}}} {\overline{V}}\big ) = 0, \qquad k = 0,1,...r. \end{aligned}$$

(E.8)

Here, we are writing ${\widetilde{\partial }} = {\widetilde{\partial }}_{{\overline{x}}}$ for the derivatives with respect to the smoothed version of ${\overline{x}}$ and similarly for ${\widetilde{{\text {div}}}}$. From these equations we see that for $k \ge 1$, there are functions $G_k,G_k,$ so that

$$\begin{aligned} h_k&= G_k(h_0, x_0 , V_0, ..., V_{k-1}),\qquad V_k = F_k(h_0, x_0, V_0, ..., V_{k-1}), \end{aligned}$$

(E.9)

using the second equation in (E.8) to replace time derivatives of ${\overline{h}}$ at $t = 0$ with a function of $V_0, V_1,..., V_{k-1}$.

We say that intial data $(V_0, h_0)$ satisfy the compatibility conditions to order r if, with the sequence $V_1, V_2,..., V_{r}$ and the functions $G_k$ defined as in (E.9), we have

$$\begin{aligned} G_k(h_0, x_0, V_0,...., V_{k-1}) \in H^1_0(\Omega ), \quad \text { for } k = 0,..., r. \end{aligned}$$

(E.10)

The significance of (E.10) is that $G_k$ must vanish on $\partial \Omega$. The point of these conditions is that they ensure that one can use a Galerkin method to solve the time-differentiated problem to find, and prove bounds for, higher-order time derivatives of h.

Provided the compatibility conditions (E.10) hold, using a Galerkin method (see [16] for a detailed proof) or duality (see [18]), one can prove that the wave equation (E.2) has a solution h with

$$\begin{aligned} D_t^k h,\,D_t^{k-1} {\widetilde{\partial }}h \in L^\infty ([0,T_0]; H^{r+1-k}(\Omega )),\quad k = 0,...,r+1, \end{aligned}$$

provided $\Vert V\Vert _{r+1,T_0} + \Vert {\widetilde{V}}\Vert _{r+1,T_0} +\Vert {\mathcal {S}}{\widetilde{x}}\Vert _{r+1, T_0} < \infty$.

The hypothesis in Theorem 1.3 is that our initial data satisfy the compatibility conditions (E.10) to order r when $\varepsilon = 0$ but in order to construct a solution for the smoothed problem we will also need initial data which satisfies the compatibility conditions to the same order with $\varepsilon > 0$. In Appendix E of [16] it was shown that this can be done under our hypotheses³ and we indicate the main points in the upcoming section E.3.

It just remains to prove the bounds (E.4)-(E.5). In fact we have already proved essentially the same bounds in section 2.6. The only substantial difference is that here we need to control normal derivatives to top order whereas in Section 2.6 we closed estimates for tangential derivatives to top order. This does not cause any serious difficulties and we sketch how to prove the needed bounds. See also [16] for a detailed proof of almost the same result.

We will just discuss how to control the highest-order part of the norm $\Vert H\Vert _{X_{T}}$ coming from the first term in the definition of the norm in (E.3). The second term in the definition of the norm is simpler to deal with. After taking one time derivative we need bounds for $\Vert \partial _y^\ell D_t^k {\widetilde{\partial }}h\Vert _{L^2(\Omega )}$ where $\ell + k = r$. If $\ell > 0$, we start by commuting $D_t^k$ with ${\widetilde{\partial }}h$. The commutator will be harmless at this point because it involves time derivatives of ${\widetilde{V}}$ which we control to higher order, and so it is enough to control $\partial _y^\ell {\widetilde{\partial }}D_t^k h$. To control this term we first use the pointwise estimate (B.1) and the elliptic estimate from Proposition B.6 for the Dirichlet problem, and so it suffices to control $\partial _y^{\ell -1} D_t^k {\widetilde{\Delta }}h$. We note that when $k = 0$ this estimate requires a bound for $\Vert {\mathcal {S}}{\widetilde{x}}\Vert _{X_{T_1}}$ which is why this quantity appears in our estimates. Writing (E.2) as

$$\begin{aligned} {\widetilde{\Delta }}h = -({\widetilde{\partial }}_i {\widetilde{V}}^j)({\widetilde{\partial }}_j V^i) + e_1 D_t^2 h, \qquad h|_{\partial \Omega } =0. \end{aligned}$$

and applying $\partial _y^{\ell -1}D_t^k$, we see that the term $\partial _y^{\ell -1} D_t^k \left( ({\widetilde{\partial }}_i {\widetilde{V}}^j)({\widetilde{\partial }}_j V^i)\right)$ is lower-order and so it is enough to control $\partial _y^{\ell -1} D_t^{k+2} h$. Now we note that the number of space derivatives falling on h has been reduced by two while the number of time derivatives falling on h has been increased by two. Repeating this argument as many times as needed, it remains to prove bounds for $\Vert D_t^r {\widetilde{\partial }}h\Vert _{L^2(\Omega )} + \Vert D_t^{r+1} h\Vert _{L^2(\Omega )}$. For this we use the estimates for the wave equation as in section 2.6.2, which requires applying $D_t^r$ to both sides of (E.2). We therefore need a bound for the term $\Vert D_t^r ({\widetilde{\partial }}_i {\widetilde{V}}^j {\widetilde{\partial }}_j V^i)\Vert _{L^2(\Omega )}$. When we encountered this term in earlier proof of the a priori bounds, we used that $D_t V = -{\widetilde{\partial }}h$ to close the estimates (see section 2.6.3) but we do not have an equation for V here. Instead we just note that this term involves time derivatives to top order and so we can control it by the first term in the definiton of the norm $\Vert \cdot \Vert _{X_{T_1}}$. This is the reason our norm involves an additional time derivative. Integrating the lower-order terms in time we get (E.4). The Lipschitz estimate (E.5) is proven in the same way.

1.2 Existence for the linear and smoothed relativistic problem

We now prove the same result for the linear relativistic problem. Fix a tangentially smooth vector field ${\widetilde{V}}$ and define ${\widetilde{x}}= {\widetilde{x}}(s,y)$ by

$$\begin{aligned} \frac{d}{ds} {\widetilde{x}}^\mu (s,y) = {\widetilde{V}}^\mu (s,y), \qquad {\widetilde{x}}^0(0,y) = 0, \quad {\widetilde{x}}^i(0,y) = y^i, \quad i = 1,2,3. \end{aligned}$$

The linear problem we consider is

$$\begin{aligned} D_s V_\mu + \frac{1}{2} {\widetilde{\partial }}_\mu \sigma = {\widetilde{\Gamma }}_{\mu \nu }^\alpha V_\alpha {\widetilde{V}}^\nu , \qquad \text { in } [0,S_1]\times \Omega , \qquad V_\mu \big |_{s = 0} = \mathring{V}_\mu , \end{aligned}$$

(E.11)

where $\sigma = \sigma [V]$ is determined by solving the wave equation

$$\begin{aligned}{} & {} e'(\sigma ) D_s^2 \sigma - \frac{1}{2} {\widetilde{{\nabla \!}}}_\nu ( {\widetilde{g}}^{\mu \nu } {\widetilde{{\nabla \!}}}_\mu \sigma ) = {\widetilde{{\nabla \!}}}_\mu {\widetilde{V}}^\nu {\widetilde{{\nabla \!}}}_\nu V^\mu \nonumber \\{} & {} \quad + {\widetilde{R}}_{\mu \nu \alpha }^\mu - e''(\sigma ) (D_s\sigma )^2\!, \quad \sigma |_{\partial \Omega } \!=\! 0, \quad \!\! \sigma |_{s = 0} \!= \!\sigma _0, \quad \!\! D_s \sigma |_{s = 0}\! = \!\sigma _1. \end{aligned}$$

(E.12)

As in the previous section, we will show that (E.11) is an ODE in a function space. The norms we work with are

$$\begin{aligned} \Vert u \Vert _{X_{S_0}} = {\sup }_{0 \le s \le S_0} \Vert u(s)\Vert _{r+1}, \end{aligned}$$

where

$$\begin{aligned} \Vert u(s)\Vert _{r+1}= & {} {\sum }_{k+\ell \le r} \Vert D_sD_s^{k} u(s)\Vert _{H^{\ell }(\Omega )} \nonumber \\{} & {} + {\sum }_{k+\ell \le r} \Vert D_s^k {\widetilde{\partial }}u(s)\Vert _{H^{\ell }(\Omega )} + {\sum }_{k+\ell \le r/2+2} \Vert \partial ^\ell D_s u(s)\Vert _{L^\infty (\Omega )}, \nonumber \\ \end{aligned}$$

(E.13)

where here $\Vert \beta \Vert _{H^\ell (\Omega )} = \sum _{\ell ' \le \ell } \Vert \partial _y^{\ell '} \beta \Vert _{L^2(\Omega )}$ where $\Vert \cdot \Vert _{L^2(\Omega )}$ is defined as in (3.13) and controls both space and time components. In section E.2.1 we prove that the map $V\mapsto \sigma [V]$ is well-defined if the compatibility conditions hold and $\Vert {\widetilde{V}}\Vert _{X_{S_1}} +\Vert {\mathcal {S}}{\widetilde{x}}\Vert _{X_{S_1}} < \infty$. With

$$\begin{aligned} \Sigma [V](s,y) = -\frac{1}{2}\int _{0}^s {\widetilde{\partial }}\sigma [V](s',y)\, ds', \end{aligned}$$

in section E.2.1 we prove the bounds

$$\begin{aligned} \Vert \Sigma [V]\Vert _{X_{S_1}}&\le C \big ( \Vert {\widetilde{V}}\Vert _{X_{S_1}}, \Vert {\mathcal {S}}{\widetilde{x}}\Vert _{X_{S_1}}, \Vert {\widetilde{g}}\Vert _r\big ) \big ( \Vert {\overline{V}}\Vert _{X_{S_1}} + S_1 \Vert V\Vert _{X_{S_1}}\big ), \end{aligned}$$

(E.14)

$$\begin{aligned} \Vert \Sigma [V_1]-\Sigma [V_2]\Vert _{X_{S_1}}&\le S_1 C \big ( \Vert {\widetilde{V}}\Vert _{X_{S_1}}, \Vert {\mathcal {S}}{\widetilde{x}}\Vert _{X_{S_1}}, \Vert V_1\Vert _{X_{S_1}}, \Vert V_2\Vert _{X_{S_1}},\Vert {\widetilde{g}}\Vert _{r+2}\big ) \Vert V_1-V_2\Vert _{X_{S_1}}. \end{aligned}$$

(E.15)

Here, $\Vert {\widetilde{g}}\Vert _{r+2}$ is defined as in (E.13). As in the previous section, this gives existence for (E.11).

Proposition E.2

(Existence for the linear relativistic problem) Fix $r \ge 10$ and suppose that the initial data $\mathring{V}, \mathring{\sigma }$ satisfies the compatibility conditions (E.17) to order $r+1$ and so that $\mathring{\rho } \ge \rho _1 > 0$ with $\mathring{\rho } = \rho |_{s = 0}$, for some constant $\rho _1> 0$. Let ${\widetilde{V}}\in X_{S_1}$ for some $S_1 > 0$. Then there is $S > 0$ so that the linear smoothed problem (E.11) has a unique solution $V \in X_{S}$ with and moreover with ${\overline{V}}$ the formal power series solution at $s = 0$ defined as in (E.16), V satisfies the bound

$$\begin{aligned} \Vert V\Vert _{X_{S}} \le C \big ( \Vert {\widetilde{V}}\Vert _{X_{S_1}}, \Vert {\widetilde{x}}\Vert _{X_{S_1}}, \Vert {\mathcal {S}}{\widetilde{x}}\Vert _{X_{S_1}}, \Vert {\widetilde{g}}\Vert _{r+2}\big ) \Vert {\overline{V}}\Vert _{X_{S}}, \end{aligned}$$

and the enthalpy satisfies

$$\begin{aligned}{} & {} \sup _{0 \le s \le S} {\sum }_{k +\ell \le r} \Vert D_s^k {\widetilde{\partial }}\sigma (s)\Vert _{H^{\ell }(\Omega )} + \Vert D_s^k D_s \sigma (s)\Vert _{H^{\ell }(\Omega )} \\{} & {} \quad \le C \Big (\sum _{k+\ell \le r}\!\!\! \Vert D_s^k {\widetilde{\partial }}\sigma (0)\Vert _{H^{\ell }(\Omega )} + \Vert D_s^{k} D_s \sigma (0)\Vert _{H^{\ell }(\Omega )} \Big ), \end{aligned}$$

with $C = C\big ( \Vert {\widetilde{V}}\Vert _{X_{S_1}}, \Vert {\mathcal {S}}{\widetilde{x}}\Vert _{X_{S_1}},\Vert {\widetilde{g}}\Vert _{r+2}\big )$. Moreover the resulting density $\rho = \rho (\sigma )$ defined by solving (1.7) satisfies $\rho (s,y) > \rho _1/2$ for $s \le S, y \in \Omega$.

It just remains to prove that $V\mapsto \sigma [V]$ is well-defined and that the bounds (E.14)-(E.15) hold.

1.2.1 The compatibility conditions and existence for the wave equation for the relativistic enthalpy

Let ${\overline{V}} = \sum _{k \ge 0} \frac{s^k}{k!} V_k, {\overline{\sigma }} = \sum _{k \ge 0} \frac{s^k}{k!} \sigma _k$ and ${\overline{x}}^i = x_0^i + s{\overline{V}}^i, {\overline{x}}^0 = s {\overline{V}}^0$ be a formal power series solution to (3.8)-(3.9) in the sense that

$$\begin{aligned} D_s^{k} \big ( D_s {\overline{V}} + (1/2){\widetilde{\partial }}{\overline{\sigma }}\big )|_{s = 0} = 0, \qquad D_s^{k} \big ( e({\overline{\sigma }}) D_s {\overline{\sigma }} + {\widetilde{{\text {div}}}} {\overline{V}}\big )|_{s = 0} = 0, \qquad k = 0,..., r, \nonumber \\ \end{aligned}$$

(E.16)

where ${\widetilde{\partial }}= {\widetilde{\partial }}_{{\overline{x}}}$ denotes differentiation with respect to the smoothed version of ${\overline{x}}$. Here, to get more uniform notation we are writing $V_0 = \mathring{V}$ for the initial velocity instead of for the time component of V. From these equations we see that there are functions $G_k, F_k$ with

$$\begin{aligned} \sigma _k = G_k(\sigma _0, x_0, V_0,..., V_{k-1}), \qquad V_k = F_k(\sigma _0, x_0, V_0, ..., V_{k-1}), \end{aligned}$$

and we say that initial data $(V_0, \sigma _0)$ satisfy the compatibility conditions to order r if we have

$$\begin{aligned} G_k(\sigma _0, x_0, V_0, ..., V_{k-1}) \in H^1_0(\Omega ), \qquad \text { for } k = 0,..., r. \end{aligned}$$

(E.17)

Here, for simplicity of notation we are ignoring the dependence on the metric and the Christoffel symbols. If the compatibility conditions hold to order r then as in the Newtonian case one can use a Galerkin method to construct a solution $\sigma$ to the wave equation (E.12) which satisfies

$$\begin{aligned} D_s^k \sigma , \, D_s^{k-1} {\widetilde{\partial }}\sigma \in L^\infty ([0,S_0]; H^{r+1-k}(\Omega )), \quad k = 0,..., r+1, \end{aligned}$$

provided $\Vert V\Vert _{r,S_0}+\Vert {\widetilde{V}}\Vert _{r,S_0} + \Vert {\mathcal {S}}{\widetilde{x}}\Vert _{r,S_0} < \infty$. The only difference with the Newtonian case is that the structure of the wave operator on the left-hand side of (E.12) is a bit less obvious. The observation which one needs is that using the formula (3.18) or equivalently the identities (1.33)- (1.34), the operator on the left-hand side of (E.12) can be decomposed into the sum of s derivatives $D_s^2$ and an operator which is elliptic when restricted to surfaces of constant s. Then the estimates which are needed to construct a solution by a Galerkin approximation follow in essentially the same way as the estimates we proved in sections 3.5.1-3.5.2. To prove the bounds (E.14)-(E.15) one argues exactly as in section E.2.1 but using the energy estimates from section 3.5.2 and the elliptic estimate from Proposition C.3.

1.3 Construction of initial data satisfying the compatibility conditions for the smoothed problem

In our main theorem we assumed that we were giving initial data which satisfies compatibility conditions for the non-smoothed problem but in our construction we need to find initial data which satisfies compatibility conditions for the smoothed-out problem which are different. In this section we sketch how to construct such data. See Proposition E.2 of [16] for a detailed proof.

We suppose that we are given vector fields $V, {\widetilde{V}}$ which are sufficiently smooth and consider

$$\begin{aligned}{} & {} D_t\big ( e_1 D_t h\big ) - {\widetilde{\Delta }} h = {\widetilde{\partial }}_i {\widetilde{V}}^j\, {\widetilde{\partial }}_j V^i, \quad \text { in } [0, t_1]\! \times \!\Omega , \quad \text {with}\quad h\big |_{ [0, t_1] \times \partial \Omega }=0,\nonumber \\{} & {} \quad \text {where}\quad {\widetilde{\Delta }}\!=\delta ^{ij}{\widetilde{\partial }}_i{\widetilde{\partial }}_j. \end{aligned}$$

(E.18)

We will just discuss the case that $e_1 > 0$ is a constant, the general case is similar.

We now fix $\varepsilon \ge 0$ and suppose that there are power series ${\overline{h}}(t, y) = \sum _{k\ge 0} t^k h_k^\varepsilon (y)/k!$, $\overline{{\widetilde{V}}}(t,y)= {\overline{V}}(t,y) = \sum _{k \ge 0} t^k V_k^\varepsilon (y)/k!$, ${\overline{{\widetilde{x}}}}(t,y) = {\overline{x}}(t,y) = \sum _{k \ge 0} t^k x^\varepsilon _k(y)/k!$ which satisfy the equation (E.18), the Euler equations (2.7) and the equations $D_t x = V$, $D_t {\widetilde{x}}= {\widetilde{V}}$ to order r at $t = 0$. With $h_1^\varepsilon$ defined by $e_1 h_1^\varepsilon = {\text {div}}V_0^\varepsilon$, we say that the initial data $(h_0^\varepsilon , h_1^\varepsilon )$ satisfies the compatibility conditions to order r if $h_k^\varepsilon \in H^1_0(\Omega ), k = 0,..., r$. The important part of this definition is the vanishing at the boundary. The statement about the power series just means that the higher-order coefficients $h_2^\varepsilon ,..., h_r^\varepsilon$ are determined from the given data $h_0^\varepsilon , h_1^\varepsilon$ by taking time derivatives of (E.18) at $t = 0$,

$$\begin{aligned} e_1 h_k^\varepsilon = {\widetilde{\Delta }} h_{k-2}^\varepsilon + F_k^\varepsilon [h_{(k-1)}^\varepsilon ], \end{aligned}$$

(E.19)

where we are evaluating the coefficients of ${\widetilde{\Delta }}$ at $t = 0$ and where we have introduced the notation $h_{(j)}^\varepsilon = (h_{-2}^\varepsilon , h_{-1}^\varepsilon , h_0^\varepsilon ,..., h_j^\varepsilon )$ with $h_{-2}^\varepsilon = x_0^\varepsilon , h_{-1}^\varepsilon = V_0^\varepsilon$, and where $F_k^\varepsilon$ depends on up to two derivatives of its arguments and is given by

$$\begin{aligned} F_k^\varepsilon [h_{(k-1)}^\varepsilon ] = D_t^{k-2} \Big (\big ( {\widetilde{\partial }}_i \overline{{\widetilde{V}}}^j {\widetilde{\partial }}_j {\overline{V}}^i\big ) +[D_t^{k-2}, {\widetilde{\Delta }}] {\overline{h}} \Big )\big |_{t = 0}. \end{aligned}$$

The parameter $\varepsilon$ enters through the definition of ${\widetilde{\partial }}$ as well as ${\widetilde{\Delta }}$. In this expression, ${\widetilde{\partial }}, {\widetilde{\Delta }}$ are defined as in (2.8) but with x replaced by ${\overline{x}}$. Using the fact that (2.7) holds at $t = 0$ one can write time derivatives of ${\overline{V}}, \overline{{\widetilde{V}}}$ and $t = 0$ in terms of the higher-order coefficients $h_0^\varepsilon ,..., h_r^\varepsilon$ and similarly one can write the time derivatives of ${\overline{{\widetilde{x}}}}^\varepsilon , {\overline{x}}^\varepsilon$ at $t = 0$ in terms of $V_0^\varepsilon , h_0^\varepsilon ,..., h_r^\varepsilon$.

The result we need is then the following.

Proposition E.3

Suppose that the initial data $(h_0, h_1)$ is such that when $\varepsilon = 0$ and with $h_k^0$ defined by (E.19), we have $h_k^0 \in H^1_0(\Omega )$ for $k = 0,..., r$. Suppose additionally that $e_1$ is sufficiently small. For $\varepsilon > 0$ sufficiently small, there is initial data $(h_0^\varepsilon , h_1^\varepsilon )$ so that with $h_k^\varepsilon$ defined by (E.19) we have $h_k^\varepsilon \in H^1_0(\Omega )$ for $k = 0,..., r$.

To prove this result we look for data of the form $(h_0^\varepsilon , h_1^\varepsilon ) = (h_0 + u_0^\varepsilon , h_1 + u_1^\varepsilon )$. Inserting this into (E.19) we see that if we define $u_k^\varepsilon$ by solving

$$\begin{aligned} {\widetilde{\Delta }} u_{k-2}^\varepsilon + G_k[u_{(k-1)}^\varepsilon ] = \kappa u_k^\varepsilon , \qquad \text { in } \Omega , \qquad u_k^\varepsilon =0, \qquad \text { on } \partial \Omega , \end{aligned}$$

where $u_{r-1}^\varepsilon = u_{r}^\varepsilon = 0$ and where $G_k$ is given by

$$\begin{aligned} G_k[u_{(k-1)}^\varepsilon ]= & {} \left( F_k^\varepsilon [h_{(k-1)} + u^\varepsilon _{(k-1)}] - F_k^\varepsilon [h_{(k-1)}]\right) \\{} & {} + \left( F_k^\varepsilon [h_{(k-1)}] - F_k[h_{(k-1)}]\right) + \big ({\widetilde{\Delta }} - \Delta \big )h_{k-2}, \end{aligned}$$

then the resulting $h_0^\varepsilon , h_1^\varepsilon$ satisfy the compatibility conditions to order r. To get back the data for $V_0^\varepsilon$ for $\varepsilon > 0$ one just takes $V_0^\varepsilon = V_0 + \nabla u_{-1}^\varepsilon$ where $\Delta u_{-1}^\varepsilon = e_1 h_1^\varepsilon$, $u_{-1}^\varepsilon = 0$ on $\partial \Omega$. The above gives a system of nonlinear elliptic equations which can be solved by iteration. Given $(u_{0}^{\varepsilon ,\nu -1}, \dots u_{r}^{\nu -1})$, construct $(u_{0}^{\varepsilon ,\nu },..., u_{r}^{\varepsilon ,\nu })$ by solving the system

$$\begin{aligned} {\widetilde{\Delta }} u_{k-2}^{\varepsilon ,\nu } + G_k[u_{(k-1)}^{\varepsilon ,\nu -1}] = e_1 u_k^{\varepsilon , \nu }, \qquad \text { in } \Omega , \qquad u_k^{\varepsilon ,\nu } =0 \qquad \text { on } \partial \Omega , \end{aligned}$$

and

$$\begin{aligned} u_{r-1}^{\varepsilon ,\nu } = u_{r}^{\varepsilon ,\nu } = 0, \qquad \text { in } \Omega . \end{aligned}$$

Provided $e_1$ is taken sufficiently small, one can use the elliptic estimates from Proposition B.6 to prove that the above sequence $(u^{\varepsilon ,\nu }_0,..., u^{\varepsilon ,\nu }_r)$ is uniformly bounded and Cauchy with respect to the norms $\sum _{k \le r} \Vert u^{\nu ,\varepsilon }_k\Vert _{H^{r-k}(\Omega )}$. See Proposition E.2 of [16] for a detailed proof.

1.4 Construction of compatible data for the relativistic problem

Data for the relativistic problem is constructed using the same steps as in the previous section. The wave equation is

$$\begin{aligned} e'({}_{\!}\sigma {}_{\!}) D_s^2 \sigma - \frac{1}{2} {\widetilde{\nabla }}_{\!\nu } ({\widetilde{g}}^{\mu \nu }{\widetilde{\nabla }}_{\!\mu } \sigma \!)= & {} {\widetilde{\nabla }}_{\!\mu } {\widetilde{V}}^\nu {\widetilde{\nabla }}_{\!\nu } V^\mu \nonumber \\{} & {} + {\widetilde{R}}_{\mu \nu \alpha }^\mu {\widetilde{V}}^\nu V^\alpha \! - e''({}_{\!}\sigma {}_{\!}) (D_s\sigma )^2\!\!, \nonumber \\{} & {} \qquad \text {in } [0, s_1]{}_{\!} \times {}_{\!}\Omega \text { with } \sigma \big |_{ [0, s_1] \times \partial \Omega }\!=\!0 \end{aligned}$$

(E.20)

The compatibility conditions for this equation are defined as in the previous section. We suppose that we are given formal power series in s, ${\overline{\sigma }}(s, y) = \sum _{k\ge 0} s^k \sigma _k^\varepsilon (y)/k!$, $\overline{{\widetilde{V}}}(s,y)= {\overline{V}}(s,y) = \sum _{k \ge 0} s^k V_k^\varepsilon (y)/k!$, ${\overline{{\widetilde{x}}}}(s,y) = {\overline{x}}(s,y) = \sum _{k \ge 0} s^k x^\varepsilon _k(y)/k!$ which satisfy the equation (E.20) to order r at $s = 0$. We can then solve for the higher-order coefficients $\sigma _2^\varepsilon ,..., \sigma _r^\varepsilon$ in terms of $\sigma _0^\varepsilon , \sigma _1^\varepsilon$ and the compatibility conditions are that the $\sigma _k^\varepsilon$ satisfy $\sigma _k^\varepsilon \in H_0^1(\Omega ), k = 0,... r$.

Simple modifications of the arguments used to prove Proposition E.2 from [16], using the elliptic estimates from Proposition C.3 in place of the elliptic estimate (5.8) from [16], can be used to prove:

Proposition E.4

Suppose that the initial data $(\sigma _0, \sigma _1)$ is such that when $\varepsilon = 0$, we have $\sigma _k^0 \in H^1_0(\Omega )$ for $k = 0,..., r$. Suppose additionally that $e_1 = e'(0)$ is sufficiently small. For $\varepsilon > 0$ sufficiently small, there is initial data $(\sigma _0^\varepsilon , \sigma _1^\varepsilon )$ so that $\sigma _k^\varepsilon \in H^1_0(\Omega )$ for $k = 0,..., r$.

Appendix F. The Galerkin method

In this section, for the sake of completeness we include a sketch of a Galerkin method which can be used to prove existence for the wave equation (1.30) for the enthalpy. We just discuss the Newtonian case, the relativistic case being similar.

Let $P_\lambda$ denote the orthogonal projection onto the space spanned by eigenfunctions of the usual Laplacian on $H^1_0(\Omega )$

$$\begin{aligned} P_\lambda f={\sum }_{\lambda _k\le \lambda } \langle f,\psi _k\rangle \psi _k, \end{aligned}$$

with eigenvalues $\le \!\lambda$. We now want to find the solution $h^\lambda$ to the equation

$$\begin{aligned} D_t\big ( e_1 D_t h^\lambda \big ) - {\widetilde{\Delta }}_\lambda h^\lambda =P_\lambda F,\quad \text { in } [0, t_1]\! \times \!\Omega , \quad \text {with}\quad h^\lambda \big |_{ \partial \Omega }=0, \end{aligned}$$

(F.1)

where ${\widetilde{\Delta }}_\lambda =P_\lambda {\widetilde{\Delta }}P_\lambda$, with initial data

$$\begin{aligned} h^\lambda \big |_{t=0}=P_\lambda h_0,\qquad D_t h^\lambda \big |_{t=0}=P_\lambda h_1, \end{aligned}$$

Here as before we have for simplicity assumed that $e_1$ is constant. This equation means that $h^\lambda$ is in the span of the eigenfunctions with eigenvalues $\lambda _k \le \lambda$:

$$\begin{aligned} h^\lambda (t,y)={\sum }_{\lambda _k\le \lambda } d^\lambda _k(t)\psi _k(y) \end{aligned}$$

and (F.1) is nothing but a system of second order ordinary differential equations for $d^\lambda _k$, obtained by taking the inner product with the eigenfunction of eigenvalues $\le \!\lambda$. Since the number of equations are the same as the number of eigenvalues this system and hence the equation has a unique solution.

Multiplying the equation by $D_t h^\lambda$ and integrating with respect to the measure dy we can remove the projections since one factor is already in the span of the eigenfunctions with eigenvalues $\lambda _k\le \lambda$:

$$\begin{aligned} \int _{\Omega } D_t h^\lambda \, D_t (e_1 D_t h^\lambda ) dy - \int _{\Omega }D_t h^\lambda \, {\widetilde{\Delta }} h^\lambda dy =\int _{\Omega }D_t h^\lambda \,F dy. \end{aligned}$$

Hence $h^\lambda$ satisfy exactly the same energy estimate as h with the exception that initial data are projected, but since the projection is bounded on the spaces we are considering it leads to the same energy bound as for h. Now, in the previous sections we mostly integrated with respect to the measure $d{\widetilde{x}}=\kappa dy$ in order that ${\widetilde{\triangle }}$ would be symmetric, however the difference just introduces a lower order term that can be controlled by the energy. Using this uniform energy bound for

$$\begin{aligned} \int _{\Omega } e_1 (D_t h^\lambda )^2 dy + \int _{\Omega }\delta ^{ij} {\widetilde{\partial }}_i h^\lambda \, {\widetilde{\partial }}_j h^\lambda dy, \end{aligned}$$

one obtains weak solutions as in [13]. The proof there is for time independent operator but can easily be modified as in [16]. Moreover by differentiating the equation with respect to t one obtains the same energy bounds for $h^\lambda$ replaced by $D_t h^\lambda$. Provided the compatibility conditions (discussed in the previous section) on the initial data hold, $D_t h^\lambda (0) \in H^1_0(\Omega )$ and this gives a solution in $H^2$ using the equation and the elliptic estimate for ${\widetilde{\triangle }} h^\lambda$. Since we have constructed our solution as a limit of eigenfunctions which vanish at the boundary and since we have uniform estimates, it follows that the compatibility conditions hold at later times.