Well-Posedness for a Moving Boundary Model of an Evaporation Front in a Porous Medium

Abstract

We consider a two-phase elliptic–parabolic moving boundary problem modelling an evaporation front in a porous medium. Our main result is a proof of short-time existence and uniqueness of strong solutions to the corresponding nonlinear evolution problem in an $L^p$-setting. It relies critically on nonstandard optimal regularity results for a linear elliptic–parabolic system with dynamic boundary condition.

Appendix

1.1 Parameter dependent elliptic problems

Remark A.1

Let $X_i$, $Y_i$, $i=0,1$ be Banach spaces, $q\in (1,\infty )$, $J=(0,T)$. Assume $Y_1\hookrightarrow Y_0$,

$$\begin{aligned}\begin{array}{rl} A\in &{} L_\infty (J,{{\mathcal {L}}}_{is}(X_1,Y_1)), \\ A^{-1}\in &{} L_\infty (J,{{\mathcal {L}}}_{is}(Y_1,X_1)),\\ K\in &{} L_q(J,{{\mathcal {L}}}(X_0,Y_1))\;\cap \; L_\infty (J,{{\mathcal {L}}}(X_0,Y_0)),\\ A+K\in &{} L_\infty (J,{{\mathcal {L}}}_{is}(X_0,Y_0)),\\ (A+K)^{-1}\in &{} L_\infty (J,{{\mathcal {L}}}_{is}(Y_0,X_0)),\\ F\in &{} L_\infty (J,Y_0)\;\cap \; L_q(J,Y_1). \end{array} \end{aligned}$$

and define $u\in L_\infty (J,X_0)$ by

$$\begin{aligned} u(t) := (A(t)+K(t))^{-1}F(t). \end{aligned}$$

Then $u(t)=A(t)^{-1}(F(t)-K(t)u(t))$ and therefore $u\in L_q(J,X_1)$, with estimate

$$\begin{aligned} \Vert u\Vert _{L_q(J,X_1)}\le & {} \Vert A^{-1}\Vert _{L_\infty (J,{{\mathcal {L}}}_{is}(Y_1,X_1))}(\Vert F\Vert _{L_q(J,Y_1)} +\Vert K\Vert _{L_q(J,{{\mathcal {L}}}(X_0,Y_1))}\Vert u\Vert _{L_\infty (J,X_0)})\\\le & {} \Vert A^{-1}\Vert _{L_\infty (J,{{\mathcal {L}}}_{is}(X_1,Y_1))}(\Vert F\Vert _{L_q(J,Y_1)}\\&+\Vert K\Vert _{L_q(J,{{\mathcal {L}}}(X_0,Y_1))} \Vert (A+K)^{-1}\Vert _{L_\infty (J,{{\mathcal {L}}}_{is}(Y_0,X_0))} \Vert F\Vert _{L_\infty (J,Y_0)}). \end{aligned}$$

Lemma A.2

Let $0<\beta<\alpha <1$, $\subset \mathbb {R}^n$ be a domain with $d := {\mathrm{diam}}D \le 1$ and $\phi \in BUC^\alpha (D)$ such that $\phi (x_0)=0$ for some $x_0 \in D$. Then

$$\begin{aligned} \Vert \phi \Vert _{BUC^{\beta }(D)}\le d^{\alpha -\beta }\Vert \phi \Vert _{BUC^{\alpha }(D)}. \end{aligned}$$

Proof

We have for $x,y \in D$

$$\begin{aligned} |\phi (x)|= & {} |\phi (x)-\phi (x_0)|\le d^\alpha \Vert \phi \Vert _{BUC^{\alpha }(D)},\\ |\phi (x)-\phi (y)|\le & {} |x-y|^\alpha \Vert \phi \Vert _{BUC^{\alpha }(D)} \le d^{\alpha -\beta }|x-y|^\beta \Vert \phi \Vert _{BUC^{\alpha }(D)}. \end{aligned}$$

$\square$

Lemma A.3

(Hölder estimates for coefficients in remainder terms).

Let

$$\begin{aligned}0<\theta _1<\theta _2<\theta _3 <\frac{1}{2}\left( 1-\frac{n+1}{p}\right) , \end{aligned}$$

$a\in Z^0(\Omega )$, $\xi _0\in \Omega$, $a(\xi _0,0)=0$, $\lambda \in (0,1)$, $\tau _0 \in J$. Let $\Omega _\lambda := \Omega _-\cap B(\xi _0,\lambda ) \ne \emptyset$, $Q_{\lambda ,\tau _0} := \Omega _\lambda \times (0,\tau _0)$. Then

$$\begin{aligned} a|_{Q_{\lambda ,\tau _0}}\in BUC^{\theta _1}((0,\tau _0),BUC^{\theta _2}(\Omega _\lambda )) \end{aligned}$$

and

$$\begin{aligned} \Vert a|_{Q_{\lambda ,\tau _0}}\Vert _{BUC^{\theta _1}((0,\tau _0),BUC^{\theta _2} (\Omega _\lambda ))} \le C(\tau _0^{\theta _2-\theta _1}+\lambda ^{2\theta _3-\theta _2}) \Vert a\Vert _{Z^0(\Omega )}. \end{aligned}$$

Proof

We rewrite

$$\begin{aligned} a(x,t)={\hat{a}}(x,t)+a(x,0),\quad x\in \Omega ,\,t\in J \end{aligned}$$

and estimate the terms on the right separately. (The second term will be interpreted both as a function on $\Omega$ and as a function on $\Omega \times J$ which is constant with respect to t.) Trace and embedding theorems yield $a(\cdot ,0)\in BUC^{2\theta _3}(\Omega )$,

$$\begin{aligned} \Vert a(\cdot ,0)\Vert _{BUC^{2\theta _3}(\Omega )}\le C\Vert a\Vert _{Z^0(\Omega )}, \end{aligned}$$

and thus by Lemma A.2

$$\begin{aligned} \Vert a(\cdot ,0)|_{\Omega _\lambda }\Vert _{BUC^{\theta _2}(\Omega _\lambda )} \le C\lambda ^{2\theta _3-\theta _2}\Vert a\Vert _{Z^0(\Omega )}. \end{aligned}$$

(A.1)

Furthermore, as $2\theta _2<1-\frac{n+1}{p}$ we have by embedding

$$\begin{aligned} {\hat{a}}\in BUC^{\theta _2}(J,BUC^{\theta _2}(\Omega )) \end{aligned}$$

and by restriction

$$\begin{aligned} {\hat{a}}|_{Q_{\lambda ,\tau _0}}\in BUC^{\theta _2}((0,\tau _0),BUC^{\theta _2}(\Omega _\lambda )) \end{aligned}$$

with estimate

$$\begin{aligned} \Vert {\hat{a}}|_{Q_{\lambda ,\tau _0}} \Vert _{BUC^{\theta _2}((0,\tau _0),BUC^{\theta _2}(\Omega _\lambda ))} \le C\Vert a\Vert _{Z^0(\Omega )}. \end{aligned}$$

Consequently, using ${\hat{a}}|_{t=0}=0$,

$$\begin{aligned} \Vert {\hat{a}}|_{Q_{\lambda ,\tau _0}} \Vert _{BUC^{\theta _1}((0,\tau _0),BUC^{\theta _2}(\Omega _\lambda ))} \le C\tau _0^{\theta _2-\theta _1}\Vert a\Vert _{Z^0(\Omega )}. \end{aligned}$$

Together with (A.1) this implies the result. $\square$

Let $a_{ij}\in Z^0(\Omega )$ be uniformly elliptic, i.e.

$$\begin{aligned} a_{ij}\xi _i\xi _j\ge \mu |\xi |^2 \hbox { in}\ \Omega \times J \end{aligned}$$

(A.2)

for some $\mu >0$. Let $b_i\in Y^-_\tau$, $f\in Y^-(\Omega _-)$, $g\in X^-_{{\mathrm{tr}}}(\Omega _-)$.

Lemma A.4

Let $M=\max \{\Vert a_{ij}\Vert _{Z^0},{\Vert b_i\Vert _{{\tilde{Y}}^-_\tau }},\mu ^{-1}\}.$ There are constants C(M), $T_0(M)$ such that for $T\le T_0$, there is precisely one solution $u\in X^-$ to the time-dependent elliptic problem

$$\begin{aligned} \left. \begin{array}{rcll} a_{ij}\partial _{ij}u+b_i\partial _i u&{}=&{}f&{}\quad \hbox {in}\,\,\Omega _-\times J,\\ u&{}=&{}g&{}\quad \hbox {on}\,\,\Gamma \times J,\\ u&{}=&{}0&{}\quad \hbox {on}\,\,\Sigma _-\times J. \end{array}\right\} \end{aligned}$$

It satisfies

$$\begin{aligned} \Vert u\Vert _{X^-(\Omega _-)}\le C(M)(\Vert f\Vert _{Y^-(\Omega _-)}+\Vert g\Vert _{X^-_{\mathrm{tr}}(\Omega _-)}) \end{aligned}$$

If $g=0$ and then .

Proof

1. We first show $u\in L_p(J,H^2_p(\Omega _-))$ with the corresponding estimate. For this we set

$$\begin{aligned} X_0:= & {} W_p^{2-\theta }(\Omega _-),\\ X_1:= & {} H^2_p(\Omega _-),\\ Y_0:= & {} W_p^{-\theta }(\Omega _-)\times W_p^{2-\theta -1/p}(\Gamma ) \times W_p^{2-\theta -1/p}(\Sigma _-),\\ Y_1:= & {} L_p(\Omega _-)\times W_p^{2-1/p}(\Gamma ) \times W_p^{2-1/p}(\Sigma _-),\\ A:= & {} (a_{ij}\partial _{ij},{\mathrm{Tr}}_\Gamma ,{\mathrm{Tr}}_\Sigma ),\\ K:= & {} (b_i\partial _i,0,0),\\ F:= & {} (f,g,0) \end{aligned}$$

and aim at the application of Remark A.1. F clearly satisfies the assumptions.

1.1. For $v\in X_0$, $t\in J$ we have

$$\begin{aligned} \Vert K(t)v\Vert _{Y_0}=\Vert b_i{(t)}\partial _iv\Vert _{W^{-\theta }_p(\Omega _-)} \le C{M}\Vert v\Vert _{X_0} \end{aligned}$$

due to the embeddings ${\tilde{Y}}^-_\tau \hookrightarrow L^\infty (J_\tau , W^{-\theta }_p(\Omega _-))$ and $X_0\hookrightarrow BUC^{1+\theta '}(\Omega _-)$ for a suitable $\theta '\in (\theta ,1)$ and the pointwise multiplicator property ([13] Theorem 3.3.2)

$$\begin{aligned} BUC^{\theta '}(\Omega _-)\cdot W_p^{-\theta }(\Omega _-)\hookrightarrow W_p^{-\theta }(\Omega _-). \end{aligned}$$

(A.3)

So $K\in L_\infty (J,{{\mathcal {L}}}(X_0,Y_0))$. Furthermore,

$$\begin{aligned}&\int _J\Vert K(t)\Vert _{{{\mathcal {L}}}(X_0,Y_1)}^p\,dt=\int _J\left( \sup _{\Vert v\Vert _{X_0}=1}\Vert b_i{(t)}\partial _i v\Vert _{L_p(\Omega _-)} \right) ^p\,dt\\&\quad =\int _J\sup _{\Vert v\Vert _{X_0}=1}\int _{\Omega _-}|b_i\partial _i v|^p\,dxdt \le C{M}^p, \end{aligned}$$

where we use the embedding $X_0\hookrightarrow BUC^1(\Omega _-)$. This shows $K\in L_p({{\mathcal {L}}}(X_0,Y_1))$.

1.2. By parallel reasonings, we get $A\in BUC(J,{{\mathcal {L}}}(X_1,Y_1))$, $A+K\in BUC(J,{{\mathcal {L}}}(X_0,Y_0))$ with norms depending only on M. The fact that for $t\in J$ we have $A(t)\in {{\mathcal {L}}}_{is}(X_1,Y_1)$, with $\Vert A(t)^{-1}\Vert _{{{\mathcal {L}}}(Y_1,X_1)}$ depending only on $\Vert a_{ij}\Vert _{Z^0(\Omega )}$ and $\mu$ follows from standard theory on elliptic boundary value problems. To get $A(t)+K(t)\in {{\mathcal {L}}}_{is}(X_0,Y_0)$ one proceeds as in the proof of [13] Theorem 4.3.3., with slight modifications due to the fact that the coefficients of A and K are not $C^\infty$. The proof remains valid, anyway, as by (A.3) and interpolation we have estimates for the lower order term of the type

$$\begin{aligned} \Vert b_i(t)\partial _iw\Vert _{W_p^{-\theta }(\Omega _-)}\le & {} C{M} \Vert w\Vert _{BUC^{1+\theta '}(\Omega _-)} \\\le & {} \varepsilon \Vert w\Vert _{X_0}+C(\varepsilon ,{M}) \Vert w\Vert _{W_p^{-\theta }(\Omega _-)} \end{aligned}$$

for any $\varepsilon >0$. Note also that we have to use (A.3) together with Lemma A.2 (with $\beta =\theta$, $\alpha =\theta '$) to estimate the highest-order error terms occurring from freezing of the coefficients.

As all assumptions of the above remark are valid, we conclude $u\in L_p(J,H^2_p(\Omega _-))$ and

$$\begin{aligned} \Vert u\Vert _{L_p(J,H^2_p(\Omega _-))}\le C(\Vert f\Vert _{Y^-(\Omega _-)}+\Vert g\Vert _{X^-_{\mathrm{tr}}(\Omega _-)}). \end{aligned}$$

2. By arguments as above, we have

$$\begin{aligned}&\int _J\int _J\frac{\Vert (K(t)-K(s))\Vert _{{{\mathcal {L}}}(X_0,Y_0)}^p}{|t-s|^{1+p\theta }}\; ds\,dt\\&\quad =\int _J\int _J\frac{\left( \sup _{\Vert v\Vert _{X_0}=1} \Vert (b_i(t)-b_i(s))\partial _iv\Vert _{W^{-\theta }_p(\Omega _-)}\right) ^p}{|t-s|^{ 1+p\theta }}\;ds\,dt\\&\quad \le C\max _i\int _J\int _J\frac{\Vert b_i(t)- b_i(s)\Vert _{W^{-\theta }_p(\Omega _-)}^p}{|t-s|^{1+p\theta }}\sup _{\Vert v\Vert _{X_0}=1} \Vert v\Vert _{BUC^{1+\theta '}(\Omega _-)}^p\;ds\,dt\le C{M}^p, \end{aligned}$$

hence $K\in W_p^{\theta }(J,{{\mathcal {L}}}(X_0,Y_0))$ and by parallel arguments $A\in W_p^{\theta }(J,{{\mathcal {L}}}(X_0,Y_0))$, with norms depending on $\Vert a_{ij}\Vert _{Z^0(\Omega )}$, $\Vert b_i\Vert _{Y^-(\Omega _-)}$. Write ${\hat{A}} := A+K$. By Step 1.2, ${\hat{A}}(0)\in {{\mathcal {L}}}_{is}(X_0.Y_0)$, and therefore there is a $\delta >0$ such that the open ball ${{\mathcal {B}}}$ in ${{\mathcal {L}}}(X_0,Y_0)$ centered at ${\hat{A}}(0)$ with radius $\delta$ lies in ${{\mathcal {L}}}_{is}(X_0,Y_0)$, and the operator $\mathop {\mathrm{inv}}$ given by $\mathop {\mathrm{inv}}(B)=B^{-1}$ is in $BUC^1({{\mathcal {B}}},{{\mathcal {L}}}(Y_0,X_0))$. Because ${\hat{A}}$ is continuous in time with values in ${{\mathcal {L}}}(X_0,Y_0)$, by shrinking J if necessary, we can arrange that ${\hat{A}}(t)\in {{\mathcal {B}}}$ for all $t\in J$. Therefore we have

$$\begin{aligned} \Vert {\hat{A}}(t)^{-1}-{\hat{A}}(s)^{-1}\Vert _{{{\mathcal {L}}}(Y_0,X_0)} \le C\Vert {\hat{A}}(t)-{\hat{A}}(s)\Vert _{{{\mathcal {L}}}(X_0,Y_0)}, \end{aligned}$$

and from this it follows straightforwardly that $\mathop {\mathrm{inv}}\circ {\hat{A}}\in W_p^\theta (J,{{\mathcal {L}}}(Y_0,X_0))$. Finally from this and $F\in W_p^\theta (J,Y_0)$ we get $u\in W_p^\theta (J,W_p^{{2}-\theta }(\Omega _-))$ and

$$\begin{aligned} \Vert u\Vert _{W_p^\theta (J,W_p^{{2}-\theta }(\Omega _-))} \le C(\Vert f\Vert _{Y^-(\Omega _-)}+\Vert g\Vert _{X^-_{\mathrm{tr}}(\Omega _-)}). \end{aligned}$$

$\square$

1.2 Uniform anisotropic embeddings, multiplications and related estimates

For the proof of our main result it is essential that the constant in the maximal regularity estimate of Theorem 3.1 is independent of T (at least for T small). However, the constants in usual embedding theorems (like $W^1_q(J)\hookrightarrow L^\infty (J)$, $q\in [1,\infty )$) are easily seen to blow up as J becomes small. The following lemma and its corollaries ensure that this does not occur when one considers only subspaces with vanishing trace(s) at $t=0$.

Lemma A.5

Let $m \in \mathbb {N}$, $q \in (1,\infty )$, $\sigma _1 = 0$ or $\sigma _1 \in (1/q, 1]$. Let further $1/q < \sigma _2 \le \cdots \le \sigma _m \le 1$ and $\sigma _1 \le \sigma _2$. Let $X_1,...,X_m$ be Banach spaces such that $X_1 \hookrightarrow X_2 \hookrightarrow \cdots \hookrightarrow X_m$. Given $T \in (0,\infty ]$, let

$$\begin{aligned} W(T) := \bigcap _{i = 1}^m W_q^{\sigma _i}((0,T),X_i), \quad W := W(\infty ). \end{aligned}$$

There exist bounded linear operators $\mathcal {E}_T: W(T) \rightarrow W$ and a constant $C > 0$ such that

$$\begin{aligned} \Vert \mathcal {E}_T u \Vert _{L_q(0,\infty ;X_i)} \le C \Vert u \Vert _{L_q(0,T;X_i)} \end{aligned}$$

for all $T \in (0,\infty ]$, $u \in L_q(0,T;X_i)$ and $i = 1,...,m$, and (in case that either $\sigma _1 \in (1/q, 1]$ or $\sigma _1 = 0$ and $m \ge 2$)

$$\begin{aligned} \Vert \mathcal {E}_T u \Vert _W \le C \Vert u \Vert _{W(T)} \end{aligned}$$

for all $T \in (0,\infty ]$ and .

Proof

For $m = 1$ and $0 < T \le T_0$ this is stated and proved in Proposition 6.1 in [10]. The case of a general $m \in \mathbb {N}$ is an immediate consequence. Moreover, a careful inspection of the proof shows that the constant C can be chosen independent of $T > T_0$, too. $\square$

Corollary A.6

Let $q \in (1,\infty )$, $\delta \ge 0$, $1/q + \delta < \sigma \le 1$ and let X be a Banach space. Given $T \in (0,\infty ]$ we have $W_q^\sigma ((0,T),X) \hookrightarrow BUC^\delta ((0,T),X)$. There exists a constant $C > 0$ such that

$$\begin{aligned} \Vert u \Vert _{BUC^\delta ((0,T),X)} \le C \Vert u \Vert _{W_q^\sigma ((0,T),X)} \end{aligned}$$

for all $T \in (0,\infty ]$ and .

Corollary A.7

Let $q \in (1,\infty )$, $2 \delta + \sigma \le 2$. Given $T \in (0,\infty ]$ we have

$$\begin{aligned} W_q^1((0,T),{L_q}(\Omega _+)) \cap L_q((0,T),W_q^2(\Omega _+)) \hookrightarrow W_q^\delta ((0,T),W_q^\sigma (\Omega _+)). \end{aligned}$$

There exists a constant $C > 0$ such that

$$\begin{aligned} \Vert u \Vert _{W_q^\delta ((0,T),W_q^\sigma (\Omega _+))} \le C \Vert u \Vert _{W_q^1((0,T),L_q(\Omega _+)) \cap L_q((0,T),W_q^2(\Omega _+))} \end{aligned}$$

for all $T \in (0,\infty ]$ and .

Let X, Y, Z be Banach spaces whose elements can be interpreted as real-valued functions on the same domain of definition. Then there is a pointwise product $(u,v)\mapsto uv$ on $X\times Y$. We write

$$\begin{aligned} X \cdot Y \hookrightarrow Z \end{aligned}$$

if for all $(u,v)\in X \times Y$ we have $uv \in Z$ and there is an $M>0$ such that

$$\begin{aligned} \Vert uv\Vert _Z \le M\Vert u\Vert _X\Vert v\Vert _Y,\quad (u,v)\in X\times Y. \end{aligned}$$

Lemma A.8

Let $q \in (1,\infty )$, $1> \rho> \sigma > 1/q$, $T > 0$ and let X, Y, Z be Banach spaces s.t. $X \cdot Y \hookrightarrow Z$. The following holds true:

• (i) $C([0,T],X) \cdot L_q(0,T;Y) \hookrightarrow L_q(0,T;Z)$ and

  $$\begin{aligned} \Vert u v \Vert _{L_q(0,T;Z)} \le M \Vert u \Vert _{C([0,T],X)} \Vert v \Vert _{L_q(0,T;Y)} \end{aligned}$$

  for all $u \in C([0,T],X)$, $v \in L_q(0,T;Y)$.

• (ii) $C^\rho ([0,T],X) \cdot W_q^\sigma ((0,T),Y) \hookrightarrow W_q^\sigma ((0,T),Z)$ and

  $$\begin{aligned} \Vert uv \Vert _{W_q^\sigma ((0,T),Z)}\le & {} C(\rho ,\sigma ,q,M) \; \big [ \Vert u \Vert _{L_\infty (0,T;X)} \Vert v \Vert _{W_q^\sigma ((0,T),Y)} \\&+ \, T^{\rho -\sigma + 1/q} \Vert v \Vert _{L_\infty (0,T;Y)} \Vert u \Vert _{C^{\rho }([0,T],X)} \big ] \end{aligned}$$

  for all $u \in C^{\rho }([0,T],X)$, $v \in W_q^\sigma ((0,T),Y)$.

• (iii) $W_q^\sigma ((0,T),X) \cdot W_q^\sigma ((0,T),Y) \hookrightarrow W_q^\sigma ((0,T),Z)$ and

  $$\begin{aligned} \Vert uv \Vert _{W_q^\sigma ((0,T),Z)}\le & {} C(q,M) \; \big [ \Vert u \Vert _{L_\infty (0,T;X)} \Vert v \Vert _{W_q^\sigma ((0,T),Y)} \\&+ \, \Vert v \Vert _{L_\infty (0,T;Y)} \Vert u \Vert _{W_q^\sigma ((0,T),X)} \big ] \end{aligned}$$

  for all $u \in W_q^\sigma ((0,T),X)$, $v \in W_q^\sigma ((0,T),Y)$.

Proof

The first statement is trivial. Let $u \in C^\rho ([0,T],X)$, $v \in W_q^\sigma ((0,T),Y)$. We have

$$\begin{aligned} \int _0^T \Vert u(t) v(t) \Vert _Z^q \; dt \le M \Vert u \Vert _{L_\infty (0,T;X)}^q \int _0^T \Vert v(t) \Vert _Y^q \; dt \end{aligned}$$

and

$$\begin{aligned} \int _0^T \int _0^T \frac{\Vert u(t)v(t) - u(s)v(s) \Vert _Z^q}{|t-s|^{1+\sigma q}} \; dt \, ds\le & {} (2M)^q \; \left[ \; \int _0^T \int _0^T \frac{\Vert u(t) \Vert _X^q \Vert v(t)-v(s) \Vert _Y^q}{|t-s|^{1+\sigma q}} \; dt \, ds \right. \nonumber \\&\left. + \; \int _0^T \int _0^T \frac{\Vert v(s) \Vert _Y^q \Vert u(t)-u(s) \Vert _X^q}{|t-s|^{1+\sigma q}} \; d(t,s) \; \right] \nonumber \\\le & {} (2M)^q \; \left[ \; \Vert u \Vert _{L_\infty (0,T;X)}^q \Vert v \Vert _{W_q^\sigma ((0,T),Y)}^q \right. \nonumber \\&\left. + \; \Vert v \Vert _{L_\infty (0,T;Y)}^q \Vert u \Vert _{C^\rho ([0,T],X)}^q \right. \nonumber \\&\left. \times \; \int _0^T \int _0^T |t-s|^{(\rho -\sigma ) q - 1} \; dt \, ds \; \right] . \end{aligned}$$

(A.4)

All our assertions follow easily from these estimates. $\square$

An immediate consequence is the following

Lemma A.9

Under the assumptions of Lemma A.8 we have

$$\begin{aligned} \Vert uv \Vert _{W_q^\sigma ((0,T),Z)}\le & {} C(\rho ,\sigma ,q,M) \; \big [ T^\rho \; \Vert u \Vert _{C^{\rho }([0,T],X)} \Vert v \Vert _{W_q^\sigma ((0,T),Y)} \\&+ \, T^{\rho +\varepsilon -\sigma +1/q} \; \Vert v \Vert _{W_q^\sigma ((0,T),Y)} \Vert u \Vert _{C^{\rho }([0,T],X)} \big ] \end{aligned}$$

for all , and $0 \le \varepsilon < \sigma - 1/q$. Moreover,

$$\begin{aligned} \Vert uv \Vert _{W_q^\sigma ((0,T),Z)} \le C(q,M) \; T^\varepsilon \; \Vert u \Vert _{W_q^\sigma ((0,T),X)} \Vert v \Vert _{W_q^\sigma ((0,T),Y)} \end{aligned}$$

for all , and $0 \le \varepsilon < \sigma -1/q$.

Remark A.10

(Product estimate, elliptic phase). Let $q \in (1,\infty )$, $N \in \mathbb {N}$ and let $D \subset \mathbb {R}^N$ be open. Lemmas A.8 and A.9 guarantee smallness of terms

$$\begin{aligned} \Vert D^2 u D v \Vert _{L_q(0,T;L_q(D)) \cap W_q^\sigma ((0,T),W_q^{-\sigma }(D))} \end{aligned}$$

for small values of T and by choosing $Y=Z=W_q^{-\sigma }(D)$, $X = W_q^{1-\sigma }(D)$ ($1/q< \sigma < 1/2$) and $Y=Z=L_q(D)$, $X = W_q^{1-\sigma }(D)$ ($1/q< \sigma < 1-N/q$), respectively.

Observe that the conditions $1/q< \sigma < 1/2$ and $1/q< \sigma < 1-N/q$ are both satisfied if $\frac{1}{q}\frac{q-1}{q-N}< \sigma < \frac{1}{2}(1-\frac{N+1}{q})$ and $q > N-1$.

Lemma A.11

Let $N \in \mathbb {N}$, $1> \sigma > 1/q$, $1 \ge r > \frac{N-1}{q-1/\sigma }$ and let U be an open set in $\mathbb {R}^{N-1}$. For $T \in (0,\infty ]$ let

$$\begin{aligned} E(T) := L_q(0,T;W_q^r(U)) \cap W_q^\sigma ((0,T),L_q(U)). \end{aligned}$$

Then $E(T) \hookrightarrow BUC((0,T) \times U)$ and E(T) is a Banach algebra. There exists a constant $C > 0$ such that

• (i) $\Vert u \Vert _\infty \le C \Vert u \Vert _{E(T)}$ for all $T \in (0,\infty ]$ and ;

• (ii) $\Vert uv \Vert _{E(T)} \le C ( \Vert u \Vert _\infty \Vert v \Vert _{E(T)} + \Vert v \Vert _\infty \Vert u \Vert _{E(T)} )$ for all $T \in (0,\infty ]$ and (where $\Vert \cdot \Vert _\infty$ denotes the sup of a function over the set $(0,T) \times M$).

  If $\delta \in [0,1]$, $r > \frac{N-1}{q(1-\delta )}$ and $\delta \sigma > 1/q + \varepsilon$, then $E(T) \hookrightarrow BUC^{\varepsilon }((0,T),BUC(U))$ and there is a constant $C > 0$ such that

• (iii) $\Vert u \Vert _\infty \le C T^{\varepsilon } \Vert u \Vert _{E(T)}$

for all $T \in (0,\infty ]$ and .

Proof

The embedding $E(T) \hookrightarrow BUC((0,T) \times U)$ is stated and proved in Lemma 4.4 in [3] and the estimate (i) follows straightforwardly from Lemma A.5, Corollary A.6. Observe that for a.e. $t \in (0,T)$

$$\begin{aligned} \Vert u(t) v(t) \Vert _{W_q^r(D)} \le C(q) \big ( \; \Vert u(t) \Vert _{C(\bar{U})} \Vert v(t) \Vert _{W_q^r(U)} + \Vert v(t) \Vert _{C(\bar{U})} \Vert u(t) \Vert _{W_q^r(U)} \; \big ), \end{aligned}$$

since $r> \frac{N-1}{q-1/\sigma } > \frac{N-1}{q}$. Hence,

$$\begin{aligned} \int _0^T \Vert u(t)v(t) \Vert _{W_q^r(U)}^q \; dt\le & {} C(q) \; \big ( \; \Vert u \Vert _\infty ^q \Vert v \Vert _{L_q(0,T;W_q^r(U))}^q \\&+\,\Vert v \Vert _\infty ^q \Vert u \Vert _{L_q(0,T;W_q^r(U))}^q \; \big ) \end{aligned}$$

and, as calculations similar to (A.4) show,

$$\begin{aligned}{}[uv]_{\sigma ,q,L_q(U)}^q \le C(q) \big ( \; \Vert u \Vert _\infty ^q \Vert v \Vert _{W_q^\sigma ((0,T),L_q(U))}^q + \Vert v \Vert _\infty ^q \Vert u \Vert _{W_q^\sigma ((0,T),L_q(U))}^q \; \big ) \end{aligned}$$

Assertion (ii) is now an easy consequence of this and of Lemma A.5, Corollary A.6. Assertion (iii) follows from Lemma 4.3 in [3] and again Lemma A.5, Corollary A.6. $\square$

Remark A.12

For $r=1-1/q$ the conditions $r > \frac{N-1}{q-1/\sigma }$, $\sigma > 1/q$ are satisfied if $\sigma > \frac{1}{q} \frac{q-1}{q-N}$. In this case, $1/(q\sigma ) < (q-N)/(q-1)$. If $\delta \in (1/(q\sigma ), (q-N)/(q-1))$, we have $1-1/q > (N-1)/(q(1-\delta ))$ and $\delta \sigma > 1/q$. Thus, (identifying $\Gamma$ with $\mathbb {R}^{n-1}$) Lemma A.11 applies to the space $Y_\theta ^B(\Gamma )$ frequently used in this paper.

1.3 Some auxiliary results concerning localizations

Let $r \in (0,1)$, $q \in [1,\infty )$, and let $\{ \Omega ^{(k)} \}_{k \in \mathcal {K}}$ be the collection of sets defined in the proof of Lemma 3.5. Suppose further that

• $f \in C^\infty (\Gamma )$, $\{ f_k \}, \{ g_k \} \subset C^\infty (\Gamma )$, $\hbox {supp}(f_k) \subset \Omega ^{(k)}$ ($k \in \mathcal {K}$);

• $\{ \psi _k \} \subset C^\infty (\Gamma )$ are such that $\hbox {supp}(\psi _k) \subset \Omega ^{(k)}$ and $\vert \partial ^\alpha \psi _k \vert _\infty \le C \lambda ^{-\alpha }$ uniformly for $k \in \mathcal {K}$.

Lemma A.13

We have

$$\begin{aligned} \Big \Vert \sum _{k \in \mathcal {K}} f_k \Big \Vert _{L_q(\Gamma )}^q \le (N_0)^q \sum _{k \in \mathcal {K}} \Vert f_k \Vert _{L_q(\Gamma )}^q \end{aligned}$$

(A.5)

and

$$\begin{aligned} \Big \Vert \sum _{k \in \mathcal {K}} f_k \Big \Vert _{W^r_q(\Gamma )}^q \le 2(2 N_0)^q \sum _{k \in \mathcal {K}} \Vert f_k \Vert _{W^r_q(\Gamma )}^q. \end{aligned}$$

(A.6)

Proof

1. Let $x \in \Gamma$. Since x is an element of at most $N_0$ of the sets $\Omega ^{(k)}$, the sum $\sum _{k \in \mathcal {K}} |f_k(x)|$ has at most $N_0$ nonzero summands. Hence

$$\begin{aligned} \left( \sum _{k \in \mathcal {K}} |f_k(x)| \right) ^q \le (N_0)^q \sum _{k \in \mathcal {K}} |f_k(x)|^q. \end{aligned}$$

2. Let $(x,y) \in \Gamma \times \Gamma$. Then the sum $\sum _{k \in \mathcal {K}} |f_k(x) - f_k(y)|$ has at most $2 N_0$ nonzero summands. Hence

$$\begin{aligned} \left( \sum _{k \in \mathcal {K}} |f_k(x) - f_k(y)| \right) ^q \le (2N_0)^q \sum _{k \in \mathcal {K}} |f_k(x) - f_k(y)|^q. \end{aligned}$$

The assertion follows from the definition of the intrinsic norms

$$\begin{aligned} \Vert f_k \Vert _{W^r_q(\Gamma )}:= & {} \Vert f_k \Vert _{L_q(\Gamma )} + [f_k]^q_{q,r,\Gamma } \\:= & {} \Vert f_k \Vert _{L_q(\Gamma )} + \int _{\Gamma } \int _{\Gamma } \frac{|f_k (x)-f_k (y)|^q}{|x-y|^{n-1+rq}} \; d\sigma (x)\;d\sigma (y). \end{aligned}$$

$\square$

Remark A.14

A special case of Lemma A.13 are the estimates

$$\begin{aligned} \Big \Vert \sum _{k \in \mathcal {K}} \psi _k f \Big \Vert _{L_q(\Gamma )}^q \le (N_0)^q \sum _{k \in \mathcal {K}} \Vert \psi _k f \Vert _{L_q(\Gamma )}^q \end{aligned}$$

(A.7)

and

$$\begin{aligned} \Big \Vert \sum _{k \in \mathcal {K}} \psi _k f \Big \Vert _{W^r_q(\Gamma )}^q \le 2(2 N_0)^q \sum _{k \in \mathcal {K}} \Vert \psi _k f \Vert _{W^r_q(\Gamma )}^q. \end{aligned}$$

(A.8)

A direct consequence of Lemma A.13 and a standard approximation argument is

Corollary A.15

Let $V_\theta ^B(\Gamma )$, $V \in \{ X,Y \}$ and p be as in Sect. 3.1. Then

$$\begin{aligned} \Big \Vert \sum _{k \in \mathcal {K}} \psi _k u_k \Big \Vert _{V_\theta ^B(\Gamma )}^p \le C(N_0,p) \sum _{k \in \mathcal {K}} \Vert \psi _k u_k \Vert _{V_\theta ^B(\Gamma )}^p \end{aligned}$$

(A.9)

for $u_k \in V_\theta ^B(\Gamma )$.

Lemma A.16

We have that

$$\begin{aligned} \sum _{k \in \mathcal {K}} \Vert \psi _k g_k \Vert ^q_{L_q(\Gamma )}\le & {} C^q \, \sum _{k \in \mathcal {K}} \Vert g_k \Vert ^q_{L_q(\Gamma )}, \end{aligned}$$

(A.10)

$$\begin{aligned} \sum _{k \in \mathcal {K}} \Vert \psi _k f \Vert ^q_{L_q(\Gamma )}\le & {} (C N_0)^q \, \Vert f \Vert ^q_{L_q(\Gamma )}, \end{aligned}$$

(A.11)

$$\begin{aligned} \sum _{k \in \mathcal {K}} [ \psi _k g_k ]^q_{q,r,\Gamma }\le & {} C^q \sum _{k \in \mathcal {K}} \Vert g_k \Vert ^q_{W^r_q(\Gamma )} + \tilde{C} \, \lambda ^{-q} \, \sum _{k \in \mathcal {K}} \Vert g_k \Vert ^q_{L_\infty (\Gamma )} \end{aligned}$$

(A.12)

and

$$\begin{aligned} \sum _{k \in \mathcal {K}} [ \psi _k f ]^q_{q,r,\Gamma }\le & {} (C N_0)^q \, \Vert f \Vert ^q_{W^r_q(\Gamma )} + \tilde{C} \, \lambda ^{-q-n+1} \, \Vert f \Vert ^q_{L_\infty (\Gamma )}. \end{aligned}$$

(A.13)

Proof

Inequality (A.10) is obvious. For (A.11) note that

$$\begin{aligned} \sum _{k \in \mathcal {K}} \sup _{x \in \Gamma } | \psi _k(x) | \le C N_0. \end{aligned}$$

This implies

$$\begin{aligned}&\sum _{k \in \mathcal {K}} \int _{\Gamma } \int _{\Gamma } \frac{|\psi _k(x)|^q |f (x) - f (y)|^q}{|x-y|^{n-1+rq}} \; d\sigma (x)\;d\sigma (y) \nonumber \\&\quad \le (C N_0)^q \, \int _{\Gamma } \int _{\Gamma } \frac{|f (x) - f (y)|^q}{|x-y|^{n-1+rq}} \; d\sigma (x)\;d\sigma (y). \end{aligned}$$

(A.14)

Inequality (A.12) follows from

$$\begin{aligned}{}[\psi _k]^q_{q,r,\Gamma }= & {} \int _{\Gamma } \int _{\Gamma } \frac{|\psi _k (x)-\psi _k (y)|^q}{|x-y|^{n-1+rq}} \; d\sigma (x)\;d\sigma (y) \nonumber \\\le & {} C^q \, \lambda ^{-q} \, \int _{\mathbb {T}^{n-1}} \int _{\mathbb {T}^{n-1}} |x-y|^{-n+1+q(1-r)} \; d\sigma (x)\;d\sigma (y) \nonumber \\\le & {} \tilde{C} \, \lambda ^{-q}, \end{aligned}$$

(A.15)

and (A.13) is obtained by combining (A.14), (A.15) and the fact that $|\mathcal {K}| \sim \lambda ^{-n+1}$. $\square$

Remark A.17

In the same way as above one obtains

$$\begin{aligned} \sum _{k \in \mathcal {K}} [ \partial _i (\psi _k g_k) ]^q_{q,r,\Gamma }\le & {} C^q \lambda ^{-q} \sum _{k \in \mathcal {K}} \Vert g_k \Vert ^q_{W^r_q(\Gamma )} + \tilde{C} \, \lambda ^{-q-1} \, \sum _{k \in \mathcal {K}} \Vert \partial _i(g_k) \Vert ^q_{L_\infty (\Gamma )} \nonumber \\&+ \; \tilde{C} \, \lambda ^{-q} \, \sum _{k \in \mathcal {K}} \Vert \partial _i g_k \Vert ^q_{L_\infty (\Gamma )} + C^q \sum _{k \in \mathcal {K}} \Vert \partial _i g_k \Vert ^q_{W^r_q(\Gamma )}, \end{aligned}$$

(A.16)

$i=1,\dots ,n-1$. From this one concludes

$$\begin{aligned} \sum _{k \in \mathcal {K}} \Vert \psi _k u_k \Vert _{X_\theta ^B(\Gamma )}^p \le C(\lambda ,p)T^\delta \sum _{k \in \mathcal {K}} \Vert u_k \Vert _{X_\theta ^B(\Gamma )}^p + C^p \sum _{k \in \mathcal {K}} \Vert u_k \Vert _{X_\theta ^B(\Gamma )}^p \end{aligned}$$

(A.17)

for , p as in Sect. 3.1, and some $\delta > 0$.