\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small Sixth Mississippi State Conference on Differential Equations and Computational Simulations, {\em Electronic Journal of Differential Equations}, Conference 15 (2007), pp. 365--375.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \setcounter{page}{365} \title[\hfilneg EJDE-2007/Conf/15\hfil Strong solutions] {Strong solutions for the Navier-Stokes equations on bounded and unbounded domains with a moving boundary} \author[J. Saal\hfil EJDE/Conf/15 \hfilneg] {J\"urgen Saal} \address{J\"urgen Saal \newline Department of Mathematics and Statistics, University of Konstanz, Box D 187, 78457 Konstanz, Germany} \email{juergen.saal@uni-konstanz.de} \thanks{Published February 28, 2007.} \subjclass[2000]{35Q30, 76D05} \keywords{Navier-Stokes equations; moving boundary; maximal regularity} \begin{abstract} It is proved under mild regularity assumptions on the data that the Navier-Stokes equations in bounded and unbounded noncylindrical regions admit a unique local-in-time strong solution. The result is based on maximal regularity estimates for the in spatial regions with a moving boundary obtained in \cite{saal2004a} and the contraction mapping principle. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{assumption}[theorem]{assumption} \newtheorem{remark}[theorem]{Remark} \section{Introduction and main result} For $T>0$ let $Q_T:=\bigcup_{t\in (0,T)}\Omega(t)\times\{t\} \subseteq \mathbb{R}^{n+1}$ be a noncylindrical space-time domain. In this note we consider the Navier-Stokes equations $$\label{NSEfo} \begin{gathered} v_t-\Delta v+(v\cdot\nabla)v+\nabla p = f \quad \mbox{in } Q_T,\\ \mathop{\rm div} v = 0 \quad \mbox{in } Q_T,\\ v = 0 \quad \mbox{on } \cup_{t\in (0,T)}\partial\Omega(t) \times\{t\},\\ v|_{t=0} = v_0 \quad \mbox{in } \Omega(0)=:\Omega_0, \end{gathered}$$ with velocity field $v$ and pressure $p$. Here we assume the moving boundary, i.e.\ the evolution of the domain $\Omega(t)$ to be determined by the level-preserving diffeomorphism $\psi: \overline{\Omega_0\times(0,T)}\to \overline{Q_T}, \quad (\xi,t)\mapsto (x,t)=\psi(\xi,t):=(\phi(\xi;t),t)$ such that for each $t\in [0,T)$, $\phi(\cdot;t)$ maps $\Omega_0$ onto $\Omega(t)$. More precisely we assume the following conditions on $\phi$ respectively $\psi$. \begin{assumption}\label{assum_phi} Let $T\in(0,\infty)$, $\Omega_0\subseteq\mathbb{R}^n$ be a domain of class $C^3$ either bounded, exterior, or a perturbed half-space. %Denote $I_T:=[0,T]$ if $T<\infty$ and $I_T:=[0,\infty)$ if $T=\infty$. Suppose that the domains $\Omega(t)$, $t\in [0,T]$, are all of the same type as $\Omega_0$, i.e.\ $\{\Omega(t)\}_{t\in [0,T]}$ is either a family of bounded domains, a family of exterior domains, or a family of perturbed half-space s. Furthermore: \begin{enumerate} \item For each $t\in [0,T]$, $\phi(\cdot;t): \overline{\Omega_0}\to\overline{\Omega(t)}$ is a $C^3$-diffeomorphism. Its inverse we denote by $\phi^{-1}(\cdot;t)$ (to emphasize that $\phi^{-1}$ is merely the inverse w.r.t. the space variables we use the semicolon notation $(\xi;t)$ for the argument of $\phi$ and $\phi^{-1}$).\label{assum_phi_1} \item For $\phi$ regarded as a function from $Q_T^0:=\Omega_0\times(0,T)$ into $\mathbb{R}^n$ we assume $\phi\in C^{3,1}_b(Q_T^0) :=\{f\in C(Q_T^0): \partial_t^k D_x^\alpha f\in C_b(Q_T^0), \ 1\le 2k+|\alpha|\le 3, k\in\mathbb{N}_0,\alpha\in\mathbb{N}_0^n\}$, where $C_b(Q_T^0)$ denotes the space of all bounded and continuous functions on $Q_T^0$.\label{assum_phi_2} \item We have $\det \nabla_\xi\phi(\xi,t)\equiv 1$, $(\xi,t)\in \overline{Q_T^0}$, (volume preserving).\label{assum_phi_3} \end{enumerate} \end{assumption} Let us remark that in view of realistic physical situations problem \eqref{NSEfo} should be considered with a certain boundary condition $v=b\neq 0$ at $\bigcup_{t\in (0,T)}\partial\Omega(t)\times\{t\}$. On the other hand, by assuming the existence of a solenoidal field $\beta$ such that $\beta=b$, the problem with $b\neq 0$ can be reduced to the case $b=0$ as described in \cite{ito61} and \cite{inouwaki77}. Therefore we restrict our considerations to the system \eqref{NSEfo} with zero boundary conditions. Also, note that in certain concrete situations the existence of the diffeormorpism $\psi$ is established. For instance in \cite{inouwaki77} the authors give as a nice example of a moving domain $\Omega(t)$ a bowl with swimming goldfishes (note that kisses are not allowed). The existence of $\psi$ in such a situation is proved in \cite{moser65} and \cite{ebinmarsden70}. Now define $\mathcal{I}^p(A):=(X,D(A))_{1-\frac1{p},p}$, for $10$, but note that under suitable additional assumptions all the assertions are still true for $T=\infty$. Employing the notation of the last section, here we are concerned with the linear problem $$\label{SEfo} \begin{gathered} v_t-\Delta v++\nabla p = f \quad \mbox{in } Q_T,\\ \mathop{\rm div} v = 0 \quad \mbox{in } Q_T,\\ v = 0 \quad \mbox{on } \cup_{t\in (0,T)}\partial\Omega(t) \times\{t\},\\ v|_{t=0} = v_0 \quad \mbox{in } \Omega(0)=:\Omega_0. \end{gathered}$$ For this system, in \cite[Theorem~2.1]{saal2004a}, the following result is proved. \begin{theorem}\label{main_result} Let $n\ge 2$, $10$ and a $C(T_0)>0$ such that $C(T)\le C(T_0)$ for all $T\le T_0$. \end{theorem} The proof of this result relies on a transform of \eqref{SEfo} via $\psi$ to a problem on the cylindrical domain $\Omega_0\times (0,T)$. The price we have to pay is that we are then left with a nonautonomous system of partial differential equations, i.e.\ the coefficients of these transformed equations depend on space and time in general. Here Assumption~\ref{assum_phi}~(2) assures that they are at least continuous. Another important point is that the transformed functions belong to the solenoidal space $L^q_\sigma(\Omega_0)$, which relies essentially on Assumption~\ref{assum_phi}~(3). More precisely this condition assures that the operator $\mathop{\rm div}$ is invariant under the chosen transform. Similar to the autonomous Stokes equations this will give us the possibility to formulate an associated abstract Cauchy problem with operators acting in $L^q_\sigma(\Omega_0)$. The idea here is to use the family of projections $P_{\Omega_0}(t):L^q(\Omega_0)\to L^q_\sigma(\Omega_0)$, which are exactly the transformed Helmholtz projections $P_{\Omega(t)}$. First let us list some obvious consequences of Assumption~\ref{assum_phi}. In view of $\det\nabla\phi(\xi,t)\equiv 1$ and $\psi(\xi,t)=(\phi(\xi;t),t)$ we also have $\det\nabla\psi=1$. Moreover, Assumption~\ref{assum_phi}~(2) implies $\psi\in C^{3,1}_b(Q_T^0;\mathbb{R}^{n+1})$. In virtue of the implicit function theorem we therefore have $\psi^{-1}\in C^{3,1}_b(Q_T;\mathbb{R}^{n+1})$ and since $\psi^{-1}(x,t)=(\phi^{-1}(x;t),t)$, $(x,t)\in Q_T$, also $\phi^{-1}\in C^{3,1}_b(Q_T;\mathbb{R}^n)$. We transform \eqref{SEfo} to a system on a fixed domain as follows. For a function $v:Q_T\to \mathbb{C}^n$ set $\tilde{v}(\xi,t):=v(\phi(\xi;t),t), \quad (\xi,t)\in\Omega_0\times [0,T].$ Then $$\label{trans_grad} (\nabla_x v)(\phi(\xi;t),t) =\left[(\nabla_\xi\phi)^{-T}\nabla_\xi\tilde{v}\right](\xi,t),$$ where $M^{-T}$ denotes $(M^T)^{-1}$ and $M^T$ stands for the transposed Matrix. Now define $$\label{spec_trans} u(\xi,t):=(\Phi(t)v)(\xi,t) :=\left[(\nabla_\xi\phi)^{-1}\tilde{v}\right](\xi,t), \quad (\xi,t)\in\Omega_0\times [0,T].$$ Assumption~\ref{assum_phi}~(1), (2), and (3) on $\phi$ imply that $\Phi(t)\in Isom(W^{k,q}(\Omega(t)),W^{k,q}(\Omega_0)) \cap Isom(W^{k,q}_0(\Omega(t)),W^{k,q}_0(\Omega_0))$ for $k=0,1,2$ and $t\in[0,T]$, and we even have the uniform estimates $$\label{uni_est_kap_phi} \|\Phi(t)v\|_{W^{k,p}(\Omega_0)} \le C_1\|v\|_{W^{k,p}(\Omega(t))} \le C_2\|\Phi(t)v\|_{W^{k,p}(\Omega_0)}$$ for all $v\in W^{k,p}(\Omega(t))$, $t\in[0,T]$, $k=0,1,2$. It is also easy to see that $\nu(x,t)$ is the outer normal at $\partial\Omega(t)$ in $x$ if and only if $\mu(\xi,t)=(\nabla\phi)^T(\xi,t)\nu(\phi(\xi,t))$ is the outer normal at $\partial\Omega_0$ in $\xi$. This implies $\nu\cdot v=0$ if and only if $\mu\cdot\Phi v=0$. Furthermore, under Assumption~\ref{assum_phi} (in particular (3)) in \cite[Proposition 2.4]{inouwaki77}\footnote{Actually in \cite{inouwaki77} only bounded $\Omega_0$ are treated. But since it is a pointwise condition the proof given there applies to each $\Omega\subset\mathbb{R}^n$.} it is proved that ${\rm div}_\xi u(\xi,t)={\rm div}_x v(\phi(\xi;t),t), \quad (\xi,t)\in \Omega_0\times[0,T].$ This implies that $\Phi(t):L^q_\sigma(\Omega(t)) \to L^q_\sigma(\Omega_0)$ is an isomorphism as well. This property of $\Phi$, which is essential in what follows, is the reason why we have to choose the special transform given in (\ref{spec_trans}). On the other hand note that this transform is responsible for the fact, that we have to assume $C^3$ boundary instead of $C^2$ only. In view of (\ref{trans_grad}) it is clear that $\Phi(t)\Delta_x\Phi(t)^{-1}$ has a representation as $$\label{rep_trans_lap} \Phi(t)\Delta_x\Phi(t)^{-1} =\sum_{|\alpha|\le 2}a_\alpha(\cdot,t)D^\alpha$$ with certain matrices $a_\alpha\in C_b^{|\alpha|,\frac{|\alpha|}{2}} (\overline{\Omega\times(0,T)})^{n\times n}$, $|\alpha|\le 2$. Explicitly we have \begin{aligned} [\Phi(t)\Delta_x\Phi(t)^{-1}u](\xi,t) &=[(\nabla_\xi\phi)^{-1}(\nabla_\xi\phi)^{-T}\nabla_\xi \cdot(\nabla_\xi\phi)^{-T}\nabla_\xi(\nabla_\xi\phi)u](\xi,t) \\ &=\sum_{i,j,k,\ell,m=1}^n\left[(\partial_{x_k}\phi^{-1}) (\partial_{x_j}\phi^{-1})^i(\partial_{x_j}\phi^{-1})^\ell\right] (\phi(\xi;t),t)\\ &\quad\times \Big[(\partial_{\xi_\ell}\partial_{\xi_i}\partial_{\xi_m} \phi^k)u^m + (\partial_{\xi_i}\partial_{\xi_m} \phi^k)\partial_{\xi_\ell}u^m \\ &\quad + (\partial_{\xi_\ell}\partial_{\xi_m} \phi^k)\partial_{\xi_i}u^m +(\partial_{\xi_m} \phi^k)\partial_{\xi_\ell}\partial_{\xi_i}u^m \Big](\xi,t). \end{aligned} \label{lap_trans} We also have \begin{aligned} \partial_t v(x,t) &=\partial_t[(\nabla_\xi\phi)u] (\phi^{-1}(x;t),t)\\ &=\sum_{i,j=1}^n(\partial_t\phi^{-1})^j(x;t) \left[(\partial_{\xi_i}\partial_{\xi_j}\phi)u^i +(\partial_{\xi_i}\phi)\partial_{\xi_j}u^i\right](\phi^{-1}(x;t),t) \\ &\quad +\sum_{i=1}^n\left[(\partial_{\xi_i}\partial_t\phi) u^i+(\partial_{\xi_i}\phi)\partial_t u^i\right](\phi^{-1}(x;t),t). \end{aligned}\label{time_der} Thus $$\label{trans_time_der} \Phi(t)\partial_t\Phi(t)^{-1} =\partial_t+\sum_{|\beta|\le 1}b_\beta(\cdot,t)D^\beta$$ with certain $b_\beta\in C_b^{2|\beta|,|\beta|} (\overline{\Omega\times(0,T)})^{n\times n}$, $|\beta|\le 1$. If we set $F:=\Phi f$ and $u_0:=\Phi(0)v_0$, as well as $\nabla^{\phi}(t) :=(\nabla_{\xi}\phi(t))^{-1}(\nabla_{\xi}\phi(t))^{-T}\nabla_{\xi}$ and $\tilde{p}:=p\circ\psi$, the transformed equations on $Q_T^0=\Omega\times(0,T)$ become $$\label{TSEFo} \begin{gathered} u_t+\sum_{|\beta|\le 1}b_\beta D^\beta u -\sum_{|\alpha|\le 2}a_\alpha D^\alpha u +\nabla^{\phi}(\cdot)\tilde{p} = F \quad \mbox{in } Q^0_T,\\ \mathop{\rm div} u = 0 \quad \mbox{in } Q^0_T,\\ u = 0 \quad \mbox{on } \partial\Omega_0\times(0,T),\\ u|_{t=0} = u_0 \quad \mbox{in } \Omega_0. \end{gathered}$$ %We call this system \eqref{TSEFo}. Since $\Phi(t)$ is an isomorphism, clearly $(u,\tilde{p})$ satisfies \eqref{TSEFo} if and only if $(v,p)$ fulfills \eqref{SEfo}. Obviously $P_{\Omega_0}(t):=\Phi(t)P_{\Omega(t)}\Phi(t)^{-1}: L^q(\Omega_0)\to L^q_\sigma(\Omega_0),\quad t\in[0,T],$ is again a projection, where $P_{\Omega(t)}$ denotes the Helmholtz projection on $L^q(\Omega(t))$. Note that $G_q(t):=(I-P_{\Omega_0}(t))L^q(\Omega_0) =\{\nabla^\phi(t) (\pi\circ \psi); \pi\in\widehat{W}^{1,q}(\Omega(t))\}.$ Thus, $P_{\Omega_0}(t)$ is not the Helmholtz projection on $L^q(\Omega_0)$ in general. As $G_q(t)$ depends on $t$ we see that also the projection $P_{\Omega_0}(t)$ does, although its range $L^q_\sigma(\Omega_0)$ is independent of $t$. Defining $$A_{\Omega_0}(t):=-P_{\Omega_0}(t) \sum_{|\alpha|\le 2}a_\alpha(\cdot,t) D^\alpha \label{def_trans_op}$$ on \begin{align*} D(A_{\Omega_0}(t)) &=\Phi(t)D(A_{\Omega(t)})\\ &=W^{2,q}(\Omega_0)\cap W^{1,q}_0(\Omega_0)\cap L^q_\sigma(\Omega_0) \\ &= D(A_{\Omega_0}), \quad t\in[0,T], \end{align*} and $$B(t):=P_{\Omega_0}(t)\sum_{|\beta|\le 1}b_\beta(\cdot,t)D^\beta, \quad t\in[0,T],\label{def_trans_pert}$$ the system \eqref{TSEFo} can be rewritten as the nonautonomous Cauchy problem $$\label{CPFu} \begin{gathered} u'(t)+(A_{\Omega_0}(t)+B(t))u(t) = F(t),\quad t\in(0,T),\\ u(0) = u_0, \end{gathered}$$ on the space $L^q_\sigma(\Omega_0)$. Observe that $A_{\Omega_0}(t)=\Phi(t)A_{\Omega(t)}\Phi(t)^{-1}$, i.e.\ it is exactly the transformed Stokes operator on $\Omega(t)$ for $t\in[0,T]$. Moreover, we see that the domain of $A_{\Omega_0}(t)$ does not depend on $t$ and equals the domain of the Stokes operator $A_{\Omega_0}$ in $L^q_\sigma(\Omega_0)$. For $T\in (0,\infty)$ and $p\in(1,\infty)$ we denote by $\mathrm{MR}_p(X,K)$ the class of all operators (and propagators) $A(\cdot)$ having maximal ($L^p$-) regularity on $X$ with a maximal regularity constant not exceeding $K$, i.e.\ there exists a unique solution $t\mapsto u(t)\in D(A(t))$ of the (eventually nonautonomous) Cauchy problem $$\label{nip} \begin{gathered} u'+A(\cdot)u = f,\quad \mbox{in } (0,T),\\ u(0)= u_0, \end{gathered}$$ satisfying the estimate $\|u'\|_{W^{1,p}((0,T);X)}+\|A(\cdot)u\|_{L^p((0,T);X)} \le K(\|f\|_{L^p((0,T);X)}+\|u_0\|_{\mathcal{I}^p(A(0))})$ for $f\in L^p((0,T);X)$ and $u_0\in \mathcal{I}^p(A(0))$. Based on two abstract results for nonautonomous systems (see \cite[Teorem~1.4 and Theorem~2.5]{saal2004a}) the following result is obtained in \cite[Theorem~3.5]{saal2004a}. \begin{proposition}\label{max_res_full_op} Let $T\in (0,\infty)$. Let $\Omega_0$, $\phi$ be as in Assumption~\ref{assum_phi} and the families $\{A_{\Omega_0}(t)\}_{t\in [0,T]}$ and $\{B(t)\}_{t\in[0,T]}$ be defined as in (\ref{def_trans_op}) and (\ref{def_trans_pert}), respectively. Then for $\mu>0$ large enough we have $$\label{equiv_pert} \|(\mu+A_{\Omega_0}(t)+B(t)) (\mu+A_{\Omega_0}(s)+B(s))^{-1}\|_{\mathcal{L}(X)}\le C, \quad t,s\in [0,T).$$ and $A_{\Omega_0}(\cdot)+B(\cdot)\in\mathrm{MR}(L^q_\sigma(\Omega_0),C(T))$. \end{proposition} We turn to the proof of the maximal regularity result for \eqref{SEfo}. \begin{proof}[Proof of Theorem \ref{main_result}] Observe that in view of \eqref{equiv_pert} and the equivalence of the norms $\|\cdot\|_{2,q}$ and $\|\cdot\|_{D(A_{\Omega_0}(0)+B(0))}$ we have $$\label{est_33} \int_0^T\left(\|u'(t)\|_q^p +\|u(t)\|_{2,q}^p\right)\mathrm{d} t \le C(T)\Big(\int_0^T\|F(t)\|_q^p\mathrm{d} t +\|u_0\|_{\mathcal{I}^p}^p\Big).$$ This yields \begin{align*} &\int_0^T\Big(\|(\partial_t+\sum_{|\beta|\le 1}b_\beta(t))u(t)\|_q^p +\|u(t)\|_{2,q}^p+\|\nabla^\phi(t)\tilde{p}(t)\|_q^p\Big)\mathrm{d} t\\ &\le C(T)\Big(\int_0^T\|F(t)\|_q^p\mathrm{d} t +\|u_0\|_{\mathcal{I}^p}^p\Big). \end{align*} for the solution $(u,p)$ of \eqref{TSEFo}. In view of (\ref{uni_est_kap_phi}), (\ref{trans_time_der}), and since $\{\Phi(t)\}_{t\in[0,T]}$ is a family of isomorphisms, this implies estimate \eqref{main_max_reg_est} for the solution of the original equations $(SE)^{\Omega(t)}_{f,v_0}$. If $v_0=0$ and $f\in L^q((0,T);L^q_\sigma(\Omega(t)))$ we may extent $f$ trivial to the interval $(0,T_0)$, where we denote the extended function by $\bar{f}$. Let $(u,p)$ and $(\bar{u},\bar{p})$ be the solution to problem \eqref{SEfo} and $(SE)^{\Omega(t)}_{\bar{f},0}$, respectively. The uniqueness of the solution implies $(\bar{u},\bar{p})|_{(0,T)}=(u,p)$. By this fact it easily follows that the constants $C(T)$ in \eqref{main_max_reg_est} can be dominated by a constant $C(T_0)$ for all $T\le T_0$. This completes the proof. \end{proof} \section{Strong solutions for the Navier-Stokes equations} \label{ns_section} Utilizing the maximal regularity for the Stokes equations, in this section we prove our main result Theorem~\ref{main_ns_result}. In order to estimate the nonlinear term in \eqref{NSEfo}, a further main ingredient in the proof will be the following embedding. \begin{lemma}\label{crucial_emb} Let $T>0$, $J=(0,T)$, $a\ge 2$, and $q>\frac{n}{a}+1$. Then we have $W^{1,q}(J;L^q(\Omega(t)))\cap L^q(J;W^{2,q}(\Omega(t))) \hookrightarrow L^{2q}(J;W^{1,aq/(a-1)}(\Omega(t))).$ If we replace $W^{1,q}(J;L^q(\Omega(t)))$ by $W_0^{1,q}(J;L^q(\Omega(t)))$ on the left hand side, then there exists a $T_0>0$ such that the embedding constant is governed by a constant $C(T_0)>0$ for all $T\le T_0$. \end{lemma} \begin{proof} Note that (\ref{uni_est_kap_phi}) and Assumption~\ref{assum_phi}~(2) imply that \begin{align*} \Phi \in& Isom(W^{\ell,p}(J;W^{k,q}(\Omega(t))), W^{\ell,p}(J;W^{k,q}(\Omega_0)))\\ & \cap\ Isom(W_0^{\ell,p}(J;W^{k,q}(\Omega(t))), W_0^{\ell,p}(J;W^{k,q}(\Omega_0))) \end{align*} for $\ell=0,1$, $k=0,1,2$, and $1\le p,q\le \infty$. In particular we have $$\label{1} \|\Phi v\|_{W^{\ell,p}(J;W^{k,q}(\Omega_0))} \le C_1 \| v\|_{W^{\ell,p}(J;W^{k,q}(\Omega(t)))} \le C_2 \|\Phi v\|_{W^{\ell,p}(J;W^{k,q}(\Omega_0))}$$ for all $v\in W^{\ell,p}(J;W^{k,q}(\Omega(t)))$, $\ell=0,1$, $k=0,1,2$, with $C_1,C_2$ independent of $T>0$. Therefore it suffices to prove the embedding $W^{1,q}(J;L^q(\Omega_0))\cap L^q(J;W^{2,q}(\Omega_0)) \hookrightarrow L^{2q}(J;W^{1,aq/(a-1)}(\Omega_0)),$ and that this embedding is even valid with an embedding constant independent of $T\le T_0$, if we assume zero time trace at $t=0$. It is a known fact that for $s\in[0,1]$, $$\label{2} W^{1,q}(J;L^q(\Omega_0))\cap L^q(J;W^{2,q}(\Omega_0)) \hookrightarrow W^{s,q}(J;W^{2(1-s),q}(\Omega_0)).$$ This follows e.g. by an application of the mixed derivative theorem \cite{sob75} (see also \cite{prusasim05}) for $J=\mathbb{R}$ and $\Omega_0=\mathbb{R}^n$. Employing suitable extension operators in space and time it can be seen that this embedding is still valid for $J=(0,T)$ and our $\Omega_0$, even with an embedding constant independent of $T\le T_0$, if we assume vanishing time trace at $t=0$ (see e.g. \cite[Proposition 6.1]{prusasim05} for the existence of such an extension operator). According to $q>\frac{n}{a}-1$ we can find an $\epsilon>0$ such that $q>\frac{n}{a}-1+\epsilon$. Now set $s:=(1+\epsilon)/2q$. Since $1-sq>0$ and $2q0$ and $aq/(a-1)\frac{n}{a}+1. Then Lemma~\ref{crucial_emb} implies$\mathbb{E}_1\hookrightarrow L^{2q}(J,L^{aq/(a-1)}(\Omega(t)))$. For$u,w\in\mathbb{E}_1we therefore deduce by applying first the H\"older and then the Sobolev inequality (in space) \begin{aligned} \|(u\cdot\nabla)w\|_{\mathbb{F}_1} &\le \|u\|_{L^{2q}(J,L^{aq}(\Omega(t)))} \|\nabla w\|_{L^{2q}(J,L^{aq/(a-1)}(\Omega(t)))} \\ &\le C\|u\|_{L^{2q}(J,W^{1,aq/(a-1)}(\Omega(t)))} \|w\|_{L^{2q}(J,W^{1,aq/(a-1)}(\Omega(t)))}. \end{aligned} \label{est_non} Observe that the above application of the Sobolev inequality requires a second condition onq$, namely that $$\label{10} q>n\frac{a-2}{a}.$$ Since relation (\ref{11}) is decreasing in$a$and (\ref{10}) is increasing in$a$, the best possible value for$q$is reached at the intersection point of the graphs of the two equations$y=\frac{n}{a}+1$and$y=n\frac{a-2}{a}$, which is $(a,y)=\Big(\frac{3n}{n-1},(n+2)/3\Big).$ Thus, by the assumption$q>(n+2)/3$and by setting$a=\frac{3n}{n-1}$the two conditions (\ref{11}) and (\ref{10}) are satisfied, which justifies the application of Lemma~\ref{crucial_emb} and the Sobolev embedding in estimate (\ref{est_non}). Now fix$T_0>0$. Let$\mathbb{B}_r(0)\subseteq\mathbb{E}_0$be the ball around$0$with radius$r$, and$(\bar{v},\bar{p})\in\ \mathbb{B}_r(0)$. Applying (\ref{est_non}) to$H_0(\bar{v},\bar{p})yields \begin{align*} \|H_0(\bar{v},\bar{p})\|_{\mathbb{F}} &\le \|F(\bar{v}+v^*)\|_{\mathbb{F}_1}\\ &\le C\left(\|\bar{v}\|^2_{a,q} +\|\bar{v}\|_{a,q} \|v^*\|_{a,q} +\|v^*\|^2_{a,q}\right), \end{align*} where\|\cdot\|_{a,q}$denotes the norm of the space$L^{2q}(J;W^{1,aq/(a-1)}(\Omega(t)))$. Applying Lemma~\ref{crucial_emb} to the terms involving$\bar{v}$results in $$\label{12} \|H_0(\bar{v},\bar{p})\|_{\mathbb{F}} \le C\left(\|(\bar{v},\bar{p})\|^2_{\mathbb{E}_0} +\|(\bar{v},\bar{p})\|_{\mathbb{E}_0} \|v^*\|_{a,q} +\|v^*\|^2_{a,q}\right)$$ for all$T\le T_0$in view of$(\bar{v},\bar{p})\in\mathbb{E}_0$. Note that by definition$H_0\in\mathcal{L}(\mathbb{E}_0,\mathbb{F}_0)$. According to Theorem~\ref{main_result} we have$\|L_T^{-1}\|_{\mathcal{L}(\mathbb{F}_0,\mathbb{E}_0)}\le C(T_0)$for all$T\le T_0$. Hence, there exists a constant$C_0>0$independent of$T\le T_0such that \begin{align*} \|L_T^{-1}H_0(\bar{v},\bar{p})\|_{\mathbb{E}_0} &\le \|L_T^{-1}\|_{\mathcal{L}(\mathbb{F}_0,\mathbb{E}_0)} \|H_0(\bar{v},\bar{p})\|_{\mathbb{F}}\\ &\le C_0\left(\|(\bar{v},\bar{p})\|^2_{\mathbb{E}_0} +\|(\bar{v},\bar{p})\|_{\mathbb{E}_0} \|v^*\|_{a,q} +\|v^*\|^2_{a,q}\right). \end{align*} Observe thatv^*$is a fixed function only depending on the data$(f,v_0)$. Hence we may choose$r>0$small so that$r<\max\{1,1/3C_0\}$and then$T>0$small such that $\|v^*\|_{a,q}<\frac{r}{3C_0}.$ This implies that $\|L_T^{-1}H_0(\bar{v},\bar{p})\|_{\mathbb{E}_0}\le r,$ that is$L_T^{-1}H_0(\mathbb{B}_r(0))\subseteq \mathbb{B}_r(0)$. To see that$L_T^{-1}H_0is a contraction observe that \begin{align*} &\|L_T^{-1}H_0(\bar{v}_1,\bar{p}_1)-L_T^{-1}H_0(\bar{v}_2,\bar{p}_2)\|_{\mathbb{E}_0}\\ &\le \|L_T^{-1}\|_{\mathcal{L}(\mathbb{F}_0,\mathbb{E}_0)} \|H_0(\bar{v},\bar{p})-H_0(\bar{v},\bar{p})\|_{\mathbb{F}_0}\\ &\le\ C(T_0)\Big(\|[(\bar{v}_1-\bar{v}_2)\cdot\nabla] v^*\|_{\mathbb{F}_1} +\|(v^*\cdot\nabla)(\bar{v}_1-\bar{v}_2) \|_{\mathbb{F}_1}\\ &\quad +\|[(\bar{v}_1-\bar{v}_2)\cdot\nabla] \bar{v}_1\|_{\mathbb{F}_1} +\|(\bar{v}_2\cdot\nabla)(\bar{v}_1-\bar{v}_2) \|_{\mathbb{F}_1} \Big). \end{align*} By applying (\ref{est_non}) and Lemma~\ref{crucial_emb} we obtain in a similar way as above that \begin{align*} &\|L_T^{-1}H_0(\bar{v}_1,\bar{p}_1)-L_T^{-1}H_0(\bar{v}_2,\bar{p}_2)\|_{\mathbb{E}_0} \\ &\le\ C_0\Big(\|v^*\|_{a,q} +\|(\bar{v}_1,\bar{p}_1)\|_{\mathbb{E}_0} +\|(\bar{v}_2,\bar{p}_2)\|_{\mathbb{E}_0}\Big) \|(\bar{v}_1-\bar{v}_2,\bar{p}_1-\bar{p}_2) \|_{\mathbb{E}_0} \end{align*} with a constantC_0>0$not depending on$T\le T_0$. 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