\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
\begin{equation} \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}
\end{equation}
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 $1<p<\infty$,
where the latter space denotes
the real interpolation space of a Banach space $X$ and the domain
$D(A)$ of a closed operator $A$ in $X$.
For $t\in[0,T)$ we denote by
\begin{gather*}
A_{\Omega(t)}=-P_{\Omega(t)}\Delta\quad\mbox{defined on}\\
D(A_{\Omega(t)})=W^{2,q}(\Omega(t))\cap W^{1,q}_0(\Omega(t))
\cap L^q_\sigma(\Omega(t))
\end{gather*}
as usual the Stokes operator  in the space of solenoidal fields
\[
L^q_\sigma(\Omega(t))=\overline{C^\infty_{c,\sigma}(\Omega(t))}^{L^q(\Omega(t))},
\]
where $C^\infty_{c,\sigma}(\Omega(t))
:=\{u\in C^\infty_c(\Omega(t)):\mathop{\rm div} u=0\}$.
Here $P_{\Omega(t)}:L^q(\Omega(t))\to L^q_\sigma(\Omega(t))$
denotes the Helmholtz projection associated to the Helmholtz decomposition
$L^q(\Omega(t))=L^q_\sigma(\Omega(t))\oplus G_q(\Omega(t))$,
where $G_q(\Omega(t))=\{\nabla p : p\in \widehat{W}^{1,q}(\Omega(t))\}$.
Note that the existence of
a compatible family $\{P_{\Omega,q}\}_{q\in(1,\infty)}$
of bounded projections
$P_\Omega=P_{\Omega,q}:L^q(\Omega)\to L^q_\sigma(\Omega)$
is well known
for all types of domains $\Omega$ considered in this note, see
e.g. \cite{fujiwara77,solonnikov77,simader92}.
For the above types of moving domains our main result is as follows.


\begin{theorem}\label{main_ns_result}
Let $n\ge 2$, $(n+2)/3<q<\infty$, and $T\in(0,\infty]$.
Let the evolution
of $\Omega(t)$, $t\in [0,T]$, be determined by a function $\psi$
satisfying Assumptions~\ref{assum_phi}.
Then, for each $v_0\in \mathcal{I}^p(A_{\Omega_0})$
and $f\in L^p((0,T);L^q(\Omega(t)))$
there exists a $T^*\in(0,T)$ and a unique solution $(v,p)$ of problem
\eqref{NSEfo}, such that
\begin{gather*}
    v\in W^{1,q}((0,T^*);L^q(\Omega(t)))
           \cap L^q((0,T^*);D(A_{\Omega(t)})),\\
    p\in L^q((0,T^*);\widehat{W}^{1,q}(\Omega(t))).
\end{gather*}
\end{theorem}

\begin{remark} \label{rmk1.3} \rm
By an inspection of the proof one will realize that the
assumption on the family $\{\Omega(t)\}_{t\in[0,T]}$, which
we intrinsically use, is not the particular geometric
shape of the domains,
but the property of having maximal regularity of the
corresponding Stokes operator  $A_{\Omega(t)}$ for each fixed $t\in[0,T]$.
Thus, Theorem~\ref{main_result} stays true for each
family $\{\Omega(t)\}_{t\in[0,T]}$ with $\Omega(t)$ of the
``same type'' for all $t\in[0,T]$ such that $A_{\Omega(t)}$
has maximal regularity. This property for instance is also
known to be valid for families of asymptotically flat layers
as examined in \cite{abels03,abels04}.
\end{remark}

Some special cases of the situation in Theorem~\ref{main_result}
are considered in several former works. First investigations
of the solvability of \eqref{NSEfo} can be found in \cite{sather63}.
However, until now
there are only existence results
available under the restrictive assumptions that $q=2$,
i.e.\ in the Hilbert space case, and that
$\{\Omega(t)\}_{t\in[0,T]}$ is a family of bounded domains.
In that situation for instance in \cite{inouwaki77} the existence
of a unique local-in-time solution of
\eqref{NSEfo}
is proved. The existence of global weak solutions in
$L^2(\Omega(t))$ for the Navier-Stokes equations
was already shown in \cite{fujsau70}
(see also \cite{bock77}).
The periodic case was obtained in \cite{morimoto71},
i.e.\ if $t\mapsto \Omega(t)$ is periodic then there exists
a periodic weak solution of the Navier-Stokes equations . A result for
more regular periodic solutions can be found in
\cite{miyatera82}.
Another existence result of local-in-time strong solutions of the
Navier-Stokes equations  in $L^2_\sigma(\Omega(t))$ for bounded $\Omega(t)$
is obtained in \cite{yamota78}. There the
authors could relax a restrictive decay condition on
the right hand side $f$ assumed in \cite{inouwaki77}, but
nevertheless they were able to construct a more regular solution.

We want to remark that the assumptions on the evolution
and regularity of $\Omega(t)$ differ in the above cited
papers. This depends mainly on the method that the authors
use in their works.
The approach presented here is closely related to the method
used in \cite{inouwaki77}. Therefore we have similar assumptions
on $\psi$ (hence also on $\Omega(t)$) as in \cite{inouwaki77}.

Theorem~\ref{main_result} generalizes the above cited
results on \eqref{NSEfo} in several directions.
Firstly, we handle $L^q$-spaces for the full scale
$(n+2)/3<q<\infty$ and arbitrary space dimension $n\ge 2$,
and not only the Hilbert space case if $n=2,3$. Moreover,
we prove the existence of strong solutions under mild
regularity assumptions on the data.
Secondly, our approach also covers
various families $\{\Omega(t)\}_{t\in[0,T)}$
of unbounded domains.

As we already mentioned, Theorem~\ref{main_result}
is based on maximal regularity estimates for the
corresponding linear Stokes equations obtained in \cite{saal2004a}.
Therefore, in Section~\ref{stokes_section}
we recall the main results
given in \cite{saal2004a} in a slightly adapted form
suitable for our purposes. Utilizing the maximal regularity
for the Stokes equations and the contraction mapping principle
in Section~\ref{ns_section} we give the proof of our main result,
the unique local-in-time strong solutions to the \eqref{NSEfo}
on noncylindrical space-time domains.

We introduce some notation used in the sequel.
By $C^k(\Omega)$ we denote the space of all $k$-times
continuously differentiable functions in an open
subset $\Omega$ of $\mathbb{R}^n$, and by $C^k_b(\Omega)$ its subspace
of $k$-times bounded continuously differentiable functions.
As usual $W^{k,q}(\Omega)$ is the
Sobolev space with norm
$\|\cdot\|_{k,q}=(\sum_{j=0}^k\|\nabla^j\cdot\|_q^q)^{1/q}$
and $L^q(\Omega)=W^{0,q}(\Omega)$ the Lebesgue space of
$q$-integrable functions. We also make use of the
homogeneous Sobolev space $\widehat{W}^{1,q}(\Omega)$, consisting
of all locally integrable functions $f$ in $\Omega$ with
$\|\nabla f\|_q<\infty$, modulo constants.
Note that we do not distinguish between scalar and
vector valued Sobolev spaces, i.e.\ we write
$L^q(\Omega)$ for $(L^q(\Omega))^n$, $W^{k,q}(\Omega)$
for $(W^{k,q}(\Omega))^n$, etc. Furthermore,
$\mathcal{L}(X,Y)$ denotes the class of all bounded operators
from $X$ to $Y$ and $Isom(X,Y)$ its subclass of isomorphisms.
If $X=Y$ we set $\mathcal{L}(X):=\mathcal{L}(X,X)$ and $Isom(X):=Isom(X,X)$.
The domain of an operator $A$ in a Banach space $X$ we denote
by $D(A)$, its range by $R(A)$, and its resolvent set by
$\rho(A)$.

\section{Maximal regularity for the Stokes equations}
\label{stokes_section}

For the readers convenience we recall here the basic steps
that lead to the maximal regularity result for the
linearized version of \eqref{NSEfo} obtained
in \cite{saal2004a}. Also note that we state them
in a slightly adapted form as we will need it in
Section~\ref{ns_section}. In this context we restrict
the statements here to the case of finite $T>0$, 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
\begin{equation} \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}
\end{equation}
For this system, in \cite[Theorem~2.1]{saal2004a}, the following
result  is proved.

\begin{theorem}\label{main_result}
Let $n\ge 2$, $1<q<\infty$, and $T\in(0,\infty)$. Let the evolution
of $\Omega(t)$, $t\in [0,T]$, be determined by a function $\psi$
satisfying Assumptions~\ref{assum_phi}. Then problem
\eqref{SEfo} has a unique solution
$t\mapsto (v(t),p(t))\in D(A_{\Omega(t)})\times \widehat{W}^{1,q}(\Omega(t))$,
$t\in [0,T]$. Furthermore, this solution satisfies
the estimate
\begin{equation}
\begin{aligned}
&\int_0^T\left[\|v_t(t)\|^p_{L^q(\Omega(t))}
+\|v(t)\|^p_{W^{2,q}(\Omega(t))}
+\|\nabla p(t)\|^p_{L^q(\Omega(t))}
\right]\mathrm{d} t\\
&\le C(T)\Big(\|v_0\|^p_{\mathcal{I}^p(A_{\Omega_0})}
+\int_0^T\|f(t)\|^p_{L^q(\Omega(t))}\mathrm{d} t\Big)
\end{aligned}\label{main_max_reg_est}
\end{equation}
for all $v_0\in \mathcal{I}^p(A_{\Omega_0})$ and $f\in L^p((0,T);L^q(\Omega(t)))$.
If $v_0=0$, then the constant $C(T)$ in \eqref{main_max_reg_est}
is uniformly bounded from above on finite intervals,
more precisely there is a $T_0>0$ 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
\begin{equation}\label{trans_grad}
(\nabla_x v)(\phi(\xi;t),t)
=\left[(\nabla_\xi\phi)^{-T}\nabla_\xi\tilde{v}\right](\xi,t),
\end{equation}
where $M^{-T}$ denotes $(M^T)^{-1}$ and $M^T$ stands for the
transposed Matrix.
Now define
\begin{equation}\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].
\end{equation}
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
\begin{equation}\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)}
\end{equation}
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
\begin{equation}\label{rep_trans_lap}
\Phi(t)\Delta_x\Phi(t)^{-1}
=\sum_{|\alpha|\le 2}a_\alpha(\cdot,t)D^\alpha
\end{equation}
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{equation}
\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}
\end{equation}
We also have
\begin{equation}
\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}
\end{equation}
Thus
\begin{equation}\label{trans_time_der}
\Phi(t)\partial_t\Phi(t)^{-1}
=\partial_t+\sum_{|\beta|\le 1}b_\beta(\cdot,t)D^\beta
\end{equation}
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
\begin{equation} \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}
\end{equation}
%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
\begin{equation}
A_{\Omega_0}(t):=-P_{\Omega_0}(t)
\sum_{|\alpha|\le 2}a_\alpha(\cdot,t) D^\alpha
\label{def_trans_op}
\end{equation}
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
\begin{equation}
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}
\end{equation}
the system \eqref{TSEFo} can be rewritten  as the
nonautonomous Cauchy problem
\begin{equation} \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}
\end{equation}
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
\begin{equation}\label{nip}
\begin{gathered}
u'+A(\cdot)u = f,\quad \mbox{in } (0,T),\\
u(0)= u_0,
\end{gathered}
\end{equation}
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
\begin{equation}\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).
\end{equation}
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
\begin{equation}\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).
\end{equation}
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
\begin{equation}\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))}
\end{equation}
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 $s\in[0,1]$,
\begin{equation}\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)).
\end{equation}
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 $2q<q/(1-sq)$ the Sobolev embedding
implies
\[
    \|v\|_{L^{2q}(J;W^{2(1-s),q}(\Omega_0))}
    \le C
    \|v\|_{W^{s,q}(J;W^{2(1-s),q}(\Omega_0))}.
\]
Furthermore, we have $n-sq>0$ and $aq/(a-1)<nq/(n-(1-2s))$.
Thus, we may apply the Sobolev embedding also in space
to the result
\begin{align*}
    \|v\|_{L^{2q}(J;W^{aq/(a-1),q}(\Omega_0))}
    &\le C\|v\|_{L^{2q}(J;W^{2(1-s),q}(\Omega_0))}\\
    &\le C
    \|v\|_{W^{s,q}(J;W^{2(1-s),q}(\Omega_0))}
\end{align*}
for $v\in W^{s,q}(J;W^{2(1-s),q}(\Omega_0))$.
In combination with (\ref{1}) and (\ref{2}) this yields
the first assertion. The additional assertion
follows by the fact that also the embedding constant of
the Sobolev embedding in time
can be chosen independently of $T\le T_0$,
if we assume vanishing time trace at $t=0$.
\end{proof}


Finally we prove our main result by employing the
contraction mapping principle.

\begin{proof}[Proof of Theorem~\ref{main_ns_result}]
First let us introduce some notation. We set
\[
    \mathbb{E}=\mathbb{E}_1\times\mathbb{E}_2,\quad \mathbb{F}=\mathbb{F}_1\times\mathbb{F}_2
\]
with
\begin{align*}
    \mathbb{E}_1&:=W^{1,q}((0,T);L^q(\Omega(t)))
             \cap L^q((0,T);D(A_{\Omega(t)})),\\
    \mathbb{E}_2&:=L^q((0,T);\widehat{W}^{1,q}(\Omega(t))),\\
    \mathbb{F}_1&:=L^p((0,T);L^q(\Omega(t))),\\
    \mathbb{F}_2&:=\mathcal{I}^p(A_{\Omega_0}).
\end{align*}
Also, denote by $\mathbb{E}_0$ and $\mathbb{F}_0$ the corresponding
spaces with vanishing time trace at $t=0$, that is
$\mathbb{E}_0=\mathbb{E}_{1,0}\times\mathbb{E}_2$ with
$\mathbb{E}_{1,0}:=W^{1,q}_0((0,T);L^q(\Omega(t)))
\cap L^q((0,T);D(A_{\Omega(t)}))$ and
$\mathbb{F}_0:=\mathbb{F}_1\times\{0\}$.
Now let $L_T$ be the solution operator of the linear problem
$(SE)^{\Omega(t)}_{\cdot,\cdot}$ and observe that
according to Theorem~\ref{main_result}
we have $L_T\in Isom(\mathbb{E},\mathbb{F})$.
Then problem \eqref{NSEfo} can formally be rephrased
as
\begin{equation}\label{rephrased}
    L_T(v,p)=(f+F(v),v_0),
\end{equation}
where $F(v):=-(v\cdot\nabla)v$.
For further purposes it will be convenient to split off
the part corresponding to the data $f$ and $v_0$. To do so
let $(v^*,p^*)$ be the solution to $(SE)^{\Omega(t)}_{f,v_0}$,
i.e.
\[
    (v^*,p^*)=L_T^{-1}(f,v_0).
\]
Moreover, we set $\bar{v}:=v-v^*$ and $\bar{p}=p-p^*$.
By this notation (\ref{rephrased}) turns into
\begin{align*}
    L_T(\bar{v},\bar{p})&=(f+F(\bar{v}+v^*),v_0)-L_T(v^*,p^*)\\
                &=(F(\bar{v}+v^*),0)=:H_0(\bar{v},\bar{p}),
\end{align*}
and therefore the fixed point equation reads as
\[
    (\bar{v},\bar{p})=L_T^{-1}H_0(\bar{v},\bar{p}),
\]
where $\bar{v}$ and $H_0(\bar{v},\bar{p})$ now have zero time trace
by construction.
Next suppose $a\ge 2$ and that
\begin{equation}\label{11}
    q>\frac{n}{a}+1.
\end{equation}
Then Lemma~\ref{crucial_emb} implies
$\mathbb{E}_1\hookrightarrow L^{2q}(J,L^{aq/(a-1)}(\Omega(t)))$.
For $u,w\in\mathbb{E}_1$ we therefore deduce by applying first the
H\"older and then the Sobolev inequality (in space)
\begin{equation}
\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}
\end{equation}
Observe that the above application of the Sobolev inequality
requires a second condition on $q$, namely that
\begin{equation}\label{10}
    q>n\frac{a-2}{a}.
\end{equation}
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
\begin{equation}\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)
\end{equation}
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 $T\le T_0$ such 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 that $v^*$ 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 $L_T^{-1}H_0$ is 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 constant $C_0>0$ not depending on $T\le T_0$.
Consequently, if we choose
$T,r>0$ such that $r,\|v^*\|_{a,q}<1/(6C_0)$,
we see that $L_T^{-1}H_0:\mathbb{B}_r(0)\to \mathbb{B}_r(0)$ is
a contraction and the assertion follows by the contraction
mapping principle.
\end{proof}

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\end{document}