The $\mathcal A$-Stokes approximation for non-stationary problems
Dominic Breit
Let $\mathcal A$ be an elliptic tensor. A function $v\in L^1(I;\LD_{\Div(B))$ is a solution to the non-stationary $\mathcal A $-Stokes problem iff \begin{align\label{abs \int_Q v\cdot\partial_tphi\dxt-\int_Q \mathcal A(\ep(v),\ep(phi))\dx=0\quad\forallphi\in C^{\infty_{0,\Div(Q), \end{align where $Q:=I\times B$, $B\subset\R^d$ bounded. If the l.h.s. is not zero but small we talk about almost solutions. We present an approximation result in the fashion of the $\mathcal A$-caloric approximation for the non-stationary $\mathcal A $-Stokes problem. Precisely, we show that every almost solution $v\in L^p(I;W^{1,p_{\Div(B))$, $1<p<\infty$, to (\ref{abs) can be approximated by a solution to (\ref{abs) in the $L^s(I;W^{1,s(B))$-sense for all $s<p$. So, we extend the stationary $\mathcal A$-Stokes approximation from \cite{BrDF to parabolic problems.
A-Stokes.tex
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\begin{document}

\begin{frontmatter}


  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
  %%% 
  %%% begin document
  %%% 
  \title{The $\mathcal A$-Stokes approximation for non-stationary problems}

  \author{Dominic Breit}
  \ead{breit@math.lmu.de}

  \address{LMU Munich, Department of Mathematics,
    Theresienstr. 39, 80333 Munich, Germany}

  %\address[ox]{Mathematical Institute, University of Oxford, 24-29 St. Giles', Oxford OX1 3LB, UK}

  \begin{abstract}
Let $\mathcal A$ be an elliptic tensor.  A function $\bfv\in L^1(I;\LD_{\Div}(B))$ is a solution to the non-stationary $\mathcal A $-Stokes problem iff
\begin{align}\label{abs}
\int_Q \bfv\cdot\partial_t\bfphi\dxt-\int_Q \mathcal A(\ep(\bfv),\ep(\bfphi))\dx=0\quad\forall\bfphi\in C^{\infty}_{0,\Div}(Q),
\end{align}
where $Q:=I\times B$, $B\subset\R^d$ bounded. If the l.h.s. is not zero but small we talk about almost solutions. We present an approximation result in the fashion of the $\mathcal A$-caloric approximation for the non-stationary $\mathcal A $-Stokes problem. Precisely, we show
that every almost solution $\bfv\in L^p(I;W^{1,p}_{\Div}(B))$, $1<p<\infty$, to (\ref{abs}) can be approximated by a solution to (\ref{abs}) in the $L^s(I;W^{1,s}(B))$-sense for all $s<p$. So, we extend the stationary $\mathcal A$-Stokes approximation from \cite{BrDF} to parabolic problems. 
  \end{abstract}

  \begin{keyword}
    solenoidal Lipschitz truncation \sep
    divergence free truncation \sep Navier-Stokes \sep Non-
    Newtonian fluids \sep A-harmonic
    approximation \sep A-caloric
    approximation\sep A-Stokes approximation\sep \MSC 35K40
    \sep 35K51 \sep 35Q35 \sep 76D03
  \end{keyword}
\end{frontmatter}


\section{Introduction}
\label{sec:intro}
A very crucial tool in the analysis of nonlinear partial differential equations is a comparison with solutions to linear problems. This powerful idea firstly appears in the work of De Giorgi \cite{DeG61} in the context of minimal surfaces. Roughly speaking we consider the nonlinear problem locally as a perturbation of a linear partial differential equation and try to transfer qualitative properties from the linear theory. Solutions to linear problems typically are smooth so we are hoping to approximate solutions to nonlinear problems by solutions to linear ones in an appropriate way in  order to establish regularity properties (of the solutions to nonlinear equations). \\
Let $\mathcal A:\R^{d\times D}\rightarrow\R^{d\times D}$ be an elliptic tensor, i.e.
\begin{align*}
\mathcal A(\bfxi,\bfxi):=\mathcal A\bfxi:\bfxi\geq\lambda|\bfxi|^2\quad\forall \bfxi\in\R^{d\times D}
\end{align*}
with $\lambda>0$ and $B\subset \R^d$ open and bounded with Lipschitz boundary (for instance a ball).
We call a function $\bfu\in W^{1,1}(\Omega)$ with
\begin{align}
\int_B \mathcal A(\nabla\bfu,\nabla\bfphi)\dx=0\quad\forall\bfphi\in C^{\infty}_0(B)
\end{align}
$\mathcal A$-harmonic on $B$. On the other, if we have for some $\delta\ll1$
\begin{align}
\Big|\int_B \mathcal A(\nabla\bfu,\nabla\bfphi)\dx\Big|\leq \delta\quad \forall\bfphi\in C^{\infty}_0(B)
\end{align}
i.e. the integral $\int_B \mathcal A(\nabla\bfu,\nabla\bfphi)\dx$ is small (compared to some norms of $\bfu$ and $\bfphi$) we call $\bfu$ almost $\mathcal A$-harmonic. De Giorgi's observation in \cite{DeG61} was the fact that almost harmonic functions can be approximated by harmonic ones. Precisely, if $\bfu\in W^{1,2}(B)$ is almost harmonic it can be approximated with respect to the $L^2(\Omega)$-norm. Since this usually does not suffice to show regularity one needs in addition a Caccioppoli inequality. It bounds the $L^2(\Omega)$-norm of the gradient in terms of the $L^2(B)$-norm of the function itself. Both together - harmonic approximation and Caccioppoli-inequality -  finally yield local $C^{1,\alpha}$-estimates.\\
Since the pioneering work of De Giorgi a lot of improvement and generalizations have been done (see \cite{DuzMin09h} for an overview). For instance the $p$-harmonic approximation was introduced in \cite{DM1} and gives a nonlinear variant considering the $p$-Laplace equation
\begin{align*}
\int_B \mathcal |\nabla\bfu|^{p-2}\nabla\bfu:\nabla\bfphi\dx=0\quad\forall\bfphi\in C^{\infty}_0(\Omega).
\end{align*}
Here almost solutions in $W^{1,p}(B)$ can be approximated by solutions in the $L^p(B)$-sense. This technique has been improved in \cite{DSV} where an approximation in $W^{1,s}(B)$ for all $s<p$ is possible. Moreover, it applies also to the more general setting of Orlicz spaces. A crucial tool for the approximation result in \cite{DSV} is the Lipschitz truncation method (originally developed in \cite{AF}; it allows to approximate a Sobolev function by a Lipschitz continuous function in a way that they are equal on a large set whose size can be controlled). Based on the $\mathcal A$-harmonic approximation and its generalizations a lot of (partial-) regularity results for nonlinear PDE's have been shown. A nice overview about regularity regularity and irregularity for elliptic problems is given in \cite{Mi}.\\

Let us turn to fluid mechanics. The $p$-Stokes problem - describing the slow stationary flow of a Non-Newtonian fluid \cite{AM},\cite{BAH} - reads as follows: for a given volume force $\bff:\Omega\rightarrow\R^d$ find $(\bfv,\pi)$ such that
\begin{align}
  \left\{\begin{array}{rlc}
      \Div\bfS(\ep(\bfv)) &=\nabla \pi-\bff& \mbox{in $B$,}\\
      \Div \bfv&=0\qquad& \mbox{in $B$,}\\
      \bfv&=\bfv_0\qquad\quad& \mbox{ \,on
        $\partial B$.}\end{array}\right.\label{1.1}
\end{align}
Here the nonlinear tensor $\bfS$ satisfies the $p$-growth condition
$$\lambda(1+|\ep|^2)^{\frac{p-2}{2}}|\bftau|^2\leq
D\bfS(\ep)(\bftau,\bftau)\leq
\Lambda(1+|\ep|^2)^{\frac{p-2}{2}}|\bftau|^2$$ for $\ep,\tau\in
\s^d$. The problem firstly appears in the mathematical literature in the work of Ladyshenskaya and Lions (see \cite{La}-\cite{La3} and \cite{Li}). Regularity results are shown in \cite{MNRR}, \cite{Fu4}, \cite{KMS},\cite{NW}, \cite{DK} and others.
In contrast to classical problems in nonlinear PDEs like the $p$-Laplace equation we have only control over the symmetric part $\ep(\bfv):=\frac{1}{2}\big(\nabla\bfv+\nabla\bfv^
T\big)$ of the gradient and more important we have the side condition $\Div\bfv=0$. A corresponding approximation theory is developed in \cite{BrDF} and approximates an almost solutions $\bfv\in W^{1,p}_{0,\Div}(B)$ to the $\mathcal A$-Stokes problem by a solution in the $W^{1,s}$-sense for all $s<p$. Let us consider the function spaces 
\begin{align*}
\LD(B)&:=\left\{\bfu\in L^1(B):\,\,\ep(\bfu)\in L^1(B)\right\},\\
\LD_{0}(B)&:=\left\{\bfu\in \LD(B):\,\,\bfu|_{\partial B}=0\right\},\\
\LD_{\Div}(B)&:=\left\{\bfu\in \LD(B):\,\,\Div\bfu=0\right\},\\
\LD_{0,\Div}(B)&:=\LD_{0}(B)\cap \LD_{\Div}(B),
\end{align*}
where $\bfu|_{\partial B}$ has to be understood in the $\mathcal H^{d-1}(\partial B)$-sense (see \cite{ts}).
A function $\bfv\in \LD_{\Div}(B)$ is a solution to the $\mathcal A $-Stokes problem iff
\begin{align}
\int_B \mathcal A(\ep(\bfv),\ep(\bfphi))\dx=0\quad\forall\bfphi\in C^{\infty}_{0,\Div}(B).
\end{align}
It is an almost solution if we have
\begin{align}
\Big|\dashint_B \mathcal A(\ep(\bfv),\ep(\bfphi))\dx\,\Big|\leq \delta\,\dashint_{B}|\ep(\bfv)|\dx\|\ep(\bfphi)\|_\infty\quad \forall\bfphi\in C^{\infty}_{0,\Div}(B)
\end{align}
with some $\delta\ll1$. Note that the formulation above is the weakest possible as it only requires $\bfv\in \LD(B)$. The approximation result in \cite{BrDF} is based on the solenoidal Lipschitz truncation combined with results developed in \cite{DLSV} for the $\mathcal A$-harmonic approximation in Orlicz spaces.\\
In order to study regularity properties of nonlinear parabolic equations Duzaar and Mingione \cite{DM05} introduce the $\mathcal A$-caloric approximation which compares almost solutions
to the $\mathcal A$-heat equation with its solutions. We call a function $\bfu\in L^1(I;W^{1,1}(B))$ with
\begin{align}
\int_Q \bfu\cdot\partial_t\bfphi\dxt-\int_Q \mathcal A(\nabla\bfu,\nabla\bfphi)\dxt=0\quad\forall\bfphi\in C^{\infty}_0(Q)
\end{align}
$\mathcal A$-caloric on $Q:=I\times B$, where $I\subset\R$ is a bounded interval. If the left hand side is small we talk about an almost $\mathcal A$-caloric function. In \cite{DM05} it is
shown that every almost $\mathcal A$-caloric function $\bfu\in L^p(I;W^{1,p}(B))$ can be approximated by a $\mathcal A$-caloric function in the parabolic $L^p$-sense. This is used to establish partial regularity results for nonlinear parabolic systems (see \cite{DuMiSt} for an overview). Despite the elliptic setting there is not so much literature available.\\
The aim of the present paper is to develop an approximation theory for non-stationary problems in fluid mechanics in the fashion of the $\mathcal A$-caloric approximation. Let us be a little bit more precise: A function $\bfv\in L^1(I;\LD_{\Div}(B))$ is a solution to the non-stationary $\mathcal A $-Stokes problem iff
\begin{align}
\int_Q \bfv\cdot\partial_t\bfphi\dxt-\int_Q \mathcal A(\ep(\bfv),\ep(\bfphi))\dx=0\quad\forall\bfphi\in C^{\infty}_{0,\Div}(Q).
\end{align}
It is an almost solution if we have
\begin{align}
\Big|\dashint_Q \bfv\cdot\partial_t\bfphi\dxt-\dashint_Q \mathcal A(\ep(\bfv),\ep(\bfphi))\dx\,\Big|\leq \delta\,\dashint_{Q}|\ep(\bfv)|\dx\|\ep(\bfphi)\|_\infty
\end{align}
for all $\bfphi\in C^{\infty}_{0,\Div}(Q)$ with some $\delta\ll1$. The main results of this paper (see Theorem \ref{Astokes} in section 4) states that every almost solution $\bfv\in L^p(I;W^{1,p}_{0,\Div}(B))$ to the non-stationary $\mathcal A$-Stokes problem can be approximated by a solution in the $L^s(I;W^{1,s}(B))$-sense for all $s<p$. So, it extend the result from \cite{BrDF} to non-stationary flows. Again we are able to work with the weakest formulation of almost solutions.
The main tool is the solenoidal Lipschitz truncation for parabolic PDE's which was recently developed in \cite{BrDS}. We present a version of it which is appropriate for our purposes in section 3. In addition to the results from \cite{BrDS} we show an approximation result for the distributional time derivative of the Lipschitz truncation (see Thm. \ref{cor:appl} d) ). In section 2 we present an $L^q$-theory for the non-stationary $\mathcal A$-Stokes problem in divergence form. It might not be surprising for experts but it is hard to find a reference in literature. \\
The investigation of the parabolic $\mathcal A$-Stokes approximation is a first step towards a partial regularity theory for nonlinear Stokes systems (results for the nonstationary $p$-Navier-Stokes are shown in \cite{Se} under certain restrictions on $p$). However the second step is still missing: an appropriate Caccioppoli-type inequality. For the heat equation (and also for parabolic systems with quadratic growth) it can be shown that solutions satisfy
\begin{align}\label{cacc}
\sup_{I_r}\dashint_{B_r}|\bfu|^2\dx+\dashint_{Q_r}|\nabla\bfu|^2\dxt\leq\,c\,\dashint_{Q_{2r}}\Big|\frac{\bfu}{r}\Big|^2\dxt.
\end{align} 
Here $Q_r=I_r\times B_r$ is a parabolic cube with radius $r$. A corresping version for parabolic systems with $p$-growth also exsists (see \cite{DuMiSt}). Due to the appearance of the pressure term it seems to be not possible to show such an estimate for Stokes systems - not even in the linear case. Some modifications of (\ref{cacc}) can be proven, but the known versions do not suffice to show partial regularity.
%As a main application we present a partial regularity theory for non-stationary power-law fluids in
%section 5 (see Theorem \ref{thm:main}). We extend the results from \cite{Se} which only allow a limited range for values for $p$. The restrictions in \cite{Se} arrise from the appearance of the pressure. By the $\mathcal A$-Stokes approxmiation and a new Caccioppoli-type inequality (see Theorem \ref{thm:cacc}) we can completely avoid the pressure term. 


\newpage


\section{$L^q$-theory for the $\mathcal A$-Stokes system}
\label{sec:reg}
The aim of this section is to present regularity results for the (non-stationary) $\mathcal A$-Stokes system depending on the right hand side (in divergence form). Let us fix for this section a bounded domain $\Omega\subset \R^d$ with $C^2$-boundary and a time interval $(0,T).$
The $\mathcal A$-Stokes problem (in the pressure-free formulation) with right hand side $\bff\in L^1(\Omega)$ reads as: find $\bfv\in \LD_{0,\Div}(B)$ such that
\begin{align}\label{eq:Af}
\int_\Omega \mathcal A(\ep(\bfv),\ep(\bfphi))\dx=\int_{\Omega} \bff\cdot\bfphi\dx\quad\text{for all }\bfphi\in C^\infty_{0,\Div}(\Omega).
\end{align}
The right hand side can also be given in divergence form, i.e.
\begin{align}\label{eq:AF}
\int_\Omega \mathcal A(\ep(\bfv),\ep(\bfphi))\dx=\int_\Omega \bfF:\nabla\bfphi\dx\quad\text{for all }\bfphi\in C^\infty_{0,\Div}(\Omega)
\end{align}
for $\bfF\in L^1(B)$. For certain purposes it is convenient to discuss the problem with a fixed divergence. To be precise for $g\in L^1_0(\Omega)$, where
$$L^q_0(\Omega):=\Big\{\bfu\in L^q(\Omega):\,\, \int_\Omega f\dx=0\Big\},$$ 
we are seeking for a function $\bfv\in \LD_{0}(B)$ with $\Div\bfv=g$ satisfying (\ref{eq:Af}) or (\ref{eq:AF}).
We have the following $L^q$-estimates.
\begin{lems}
\label{lems:Aq}
\begin{enumerate}
Let $\Omega\subset\R^d$ be a bounded $C^2$-domain and $1<q<\infty$.
\item[a)] Let $\bff\in L^q(\Omega)$ and $g\in W^{1,q}(\Omega)$ with $\int_\Omega g\dx=0$. Then there is a unique solution $\bfw\in W^{2,q}\cap W^{1,q}_{0}(\Omega)$ to (\ref{eq:Af}) such that $\Div\bfv=g$ and 
\begin{align*}
\dashint_{B}|\nabla^2\bfw|^q\dx\leq \,c\,\dashint_{\Omega}|\bff|^q\dx+c\,\dashint_{\Omega}|\nabla g|^q\dx,
\end{align*}
where $c$ only depends on $\mathcal A$ and $q$.
\item[b)] Let $\bfF\in L^q(\Omega)$ and $g\in L^q_0(\Omega)$. Then there is a unique solution $\bfw\in W^{1,q}_{0}(B)$ to (\ref{eq:AF}) such that $\Div\bfv=g$ and 
\begin{align*}
\dashint_{\Omega}|\nabla\bfw|^q\dx\leq \,c\,\dashint_{\Omega}|\bfF|^q\dx+c\,\dashint_{\Omega}|g|^q\dx,
\end{align*}
where $c$ only depends on $\mathcal A$ and $q$.
\end{enumerate}
\end{lems}
In case $\mathcal A=I$ both parts follow from \cite{AmGi}, Thm 4.1. However, the main tool in \cite{AmGi} is the theory from
\cite{ADN1,ADN2} where very general linear systems are investigated. Hence it is clear that the results also hold in case of an arbitrary
elliptic tensor $\mathcal A$.

Now we turn to the parabolic problem and the first result is a local $L^q$-estimate for weak solutions. In case of the $\mathcal A$-heat system this follows from the continuity of the corresponding semigroup (see \cite{Sh}). It is also non for the non-stationary Stokes-system (see \cite{So} and \cite{Gi}) but not in our setting.
\begin{them}
\label{thm:Lqlocal}
Let $\bff\in L^q(Q_0)$ for some $q>2$, where $Q_0:=(0,T)\times\R^d$ and let $\bfv\in L^1((0,T);\LD_{\Div}(\R^d))$ be a weak solution to
\begin{align}
\int_{Q_0} \bfv\cdot\partial_t\bfphi\dxt-\int_{Q_0} \mathcal A(\ep(\bfv),\ep(\bfphi))\dx=\int_{Q_0}\bff\cdot\bfphi\dxt
\end{align}
for all $\bfphi\in C^{\infty}_{0,\Div}([0,T)\times \R^d)$. Then we have $\nabla^2\bfv\in L^q_{loc}(Q_0)$ and there holds
\begin{align*}
\int_0^T\int_{B}|\nabla^2\bfv|^q\dxt\leq\,c\, \int_{Q_0}|\bff|^q\dxt
\end{align*}
for all balls $B\subset\R^d$.
\end{them}
\begin{proof}
The main ingredient is the proof of the following auxiliary result which has been used in a similar version
 in \cite{ByWa}.\\
We say that $Q' = I' \times B' \subset \setR \times
\Rdr$ is a parabolic cylinder if $r_{I'} =
r_{B'}^2$. For $\kappa>0$ we define the scaled cylinder $\kappa Q' :=
(\kappa I') \times (\kappa B')$.  By $\mathcal{Q}$ we denote
the set of all parabolic cylinders.  We define the
parabolic maximal operators $\mathcal M$ and $\mathcal M_s$ for $s \in
[1,\infty)$ by
%\marginpar{D: $\Ma_s$ f{\"u}r $s>1$ brauchen wir nicht mehr, oder?}
\begin{align*}
  (\mathcal M f)(t,x) &:= \sup_{ Q' \in
    \mathcal{Q}\,:\,(t,x) \in Q'} \dashint_{Q'} \abs{f(\tau,y)} \dtau \dy,
  \\
  \mathcal M_s f(t,x) &:= \big(\mathcal M(\abs{f}^s)(t,x)\big)^{\frac 1s}
\end{align*}
It is standard\cite{Ste93} that for all $q \in (s,\infty]$
\begin{align}
  \label{eq:Mascont}
  \norm{\Ma_s f}_{L^q(\R^{n+1})} &\leq c\, \norm{f}_{L^q(\R^{n+1})}.
\end{align}
\begin{itemize}
\item[i)] We start with interior estimates. Let $Q_r:=Q_r(x_0,t_0):=(t_0-r^2,t_0+r^2)\times B_r(x_0)$ be a parabolic cube such that $4Q_r\subset Q_0$ then there holds: there is a constant $N_1>0$ such that for every $\varepsilon>0$ there is $\delta=\delta(\varepsilon)>0$ such that\footnote{Since $\bff\in L^2(0,T;L^2_{loc}(\R^d))$ the standard interior regularity theory implies $\nabla^2\bfv\in L^2(0,T;L^2_{loc}(\R^d))$ and  $\partial_t\bfv\in L^2(0,T;L^2_{loc}(\R^d))$.}
\begin{align}\label{imp1}
\begin{aligned}
&\mathcal L^{d+1}\big(Q_r\cap \set{\mathcal M (|\bff|)^2>N_1^2}\big)\geq \varepsilon\mathcal L^{d+1}(Q_r)\\
\Rightarrow&\,\, Q_r\subset \set{\mathcal M(|\nabla^2\bfv|^2)>1}\cap \set{\mathcal M(|\bff|^2)>\delta^2}.
\end{aligned}
\end{align}
Let us assume for simpicity that $r=1$. In fact, we will establish (\ref{imp1}) by showing 
\begin{align}\label{imp2}
\begin{aligned}
&Q_1\cap \set{\mathcal M(|\nabla^2\bfv|^2)\leq 1}\cap \set{\mathcal M(|\bff|^2)\leq \delta^2}\neq\emptyset\\
\Rightarrow&\,\,\mathcal L^{d+1}\big(Q_1\cap \set{\mathcal M (|\nabla\bfv|)^2>N_1^2}\big)< \varepsilon\mathcal L^{d+1}(Q_1)
\end{aligned}
\end{align}
and applying a simple scaling argument.
In order to show (\ref{imp2}) we compare $\bfv$ with a solution to a homogeneous problem on $Q_4=(t_0^4,t_0^4+8^2)\times B_4(x_0^4)\subset Q_0$ (with the same boundary data) which is smooth in the interior. So let us define $\bfh$ as the unique solution to
\begin{align}
  \left\{\begin{array}{rlc}
     \partial_t\bfh- \Div\mathcal A(\ep(\bfh)) &=\nabla \pi_{\bfh}& \mbox{in $Q_4$,}\\
      \Div \bfh&=0\qquad& \mbox{in $Q_4$,}\\
      \bfh&=\bfv\qquad\quad& \mbox{ \,on
        $I_4\times\partial B_4$,}\\
\bfh(t_0^4,\cdot)&=\bfv(t_0^{4},\cdot)\qquad\quad& \mbox{ \,in
        $B_4$.}
\end{array}\right.\label{eq:h}
\end{align}
We test the difference of both equations with $\bfv-\bfh$. This yields by the ellipticity of $\mathcal A$
\begin{align*}
\sup_{t\in I_4}\int_{B_4}|\bfv(t)-\bfh(t)|^2\dx&+\int_{Q_4}|\ep(\bfv)-\ep(\bfh)|^2\dxt\\&\leq \,c\,\int_{Q_4}|\bff|^2\dxt+\,c\,\int_{Q_4}|\bfv-\bfh|^2\dxt.
\end{align*}
An application of Korn's inequality and Gronwall's lemma implies
\begin{align}\label{eq:cacc1}
\sup_{t\in I_4}\int_{B_4}|\bfv(t)-\bfh(t)|^2\dx+\int_{Q_4}|\nabla\bfv-\nabla\bfh|^2\dxt\leq \,c\,\int_{Q_4}|\bff|^2\dxt.
\end{align}
Now we choose a cut off function $\eta\in C^\infty_0(B_4)$ with $0\leq\eta\leq1$ and $\eta\equiv1$ on $B_3$. We test the equation for $\bfv-\bfh$ with $\curl(\eta^2\curl(\bfh-\bfh))$; cf. \cite{DK}. We gain
\begin{align*}
\sup_{t\in I_4}\int_{B_4}&\eta^2|\curl(\bfv-\bfh)|^2\dx+\int_{Q_4}\eta^2|\nabla\ep(\bfv)-\nabla\ep(\bfh)|^2\dxt\\&\leq \,c\,\int_{Q_4}\bff\cdot\curl(\eta^2\curl(\bfh-\bfh))\dxt\\
&+\,c(\nabla\eta)\,\int_{Q_4}|\nabla\bfv-\nabla\bfh|^2\dxt.
\end{align*}
Estimating the term involving $\bff$ by
\begin{align*}
\int_{Q_4}&\bff\cdot\curl(\eta^2\curl(\bfv-\bfh))\dxt\\&\leq\,\,c(\kappa)\int_{Q_4}|\bff|^2\dxt+\kappa\int_{Q_4}\eta^2|\nabla^2\bfv-\nabla^2\bfh|^2\dxt+c(\nabla\eta)\,\int_{Q_4}|\nabla\bfv-\nabla\bfh|^2\dxt,
\end{align*}
where $\kappa>0$ is arbitrary and using the inequality $|\nabla^2\bfu|\leq \,c\,|\nabla\ep(\bfu)|$ as well as (\ref{eq:cacc1}) shows
\begin{align}\label{eq:cacc2}
\sup_{t\in I_4}\int_{B_3}|\curl(\bfv-\bfh)(t)|^2\dx+\int_{I_4}\int_{B_3}|\nabla^2\bfv-\nabla^2\bfh|^2\dxt\leq \,c\,\int_{Q_4}|\bff|^2\dxt.
\end{align}
Now, let us assume that (\ref{imp2})$_1$ holds. Then there is a point $(t_0,x_0)\in Q_1$ such that
\begin{align}\label{eq:2.28}
\dashint_{Q_r(t_0,x_0)}|\nabla^2\bfv|^2\dxt\leq 1,\quad \dashint_{Q_r(t_0,x_0)}|\bff|^2\dxt\leq \delta^2\quad \forall r>0.
\end{align}
Since $Q_4\subset Q_6(t_0,x_0)$ we have
\begin{align}\label{eq:3.29}
\int_{Q_4}|\nabla^2\bfv|^2\dxt\leq c,\quad \int_{Q_4}|\bff|^2\dxt\leq \,c\delta^2.
\end{align}
As $\bfh$ is smooth we know that 
\begin{align}\label{eq:3.33}
N_0^2:=\sup_{Q_3}|\nabla^2\bfh|^2<\infty
\end{align}
 and can conclude that
\begin{align}\label{eq:3.34}
Q_1\cap \set{\mathcal M(|\nabla^2\bfv|^2)>N_1^2}\subset Q_1\cap \set{\mathcal M(\chi_{Q_3}|\nabla^2\bfv-\nabla^2\bfh|^2)>N_0^2}
\end{align}
if we set $N_1^2:=\max\set{4N_0^2,2^{d+2}}$. To establish (\ref{eq:3.34}) suppose that
\begin{align}\label{eq:3.35}
(t,x)\in Q_1\cap \set{\mathcal M(\chi_{Q_3}|\nabla^2\bfv-\nabla^2\bfh|^2)\leq N_0^2}.
\end{align}
If $r\leq 2$ we have $Q_{r}(t,x)\subset Q_3$ and gain by (\ref{eq:3.33})
\begin{align*}
\dashint_{Q_r(t,x)}&|\nabla^2\bfv|^2\dxt\\&\leq 2\,\dashint_{Q_r(t,x)}\chi_{Q_3}|\nabla^2\bfv-\nabla^2\bfh|^2\dxt+2\,\dashint_{Q_r(t,x)}\chi_{Q_3}|\nabla^2\bfh|^2\dxt\\
&\leq\,4N_0^2.
\end{align*}
If $r\geq2$ we have by (\ref{eq:2.28})
\begin{align*}
\dashint_{Q_r(t,x)}&|\nabla^2\bfv|^2\dxt\leq 2^{d+2}\dashint_{Q_{2r}(t_0,x_0)}|\nabla^2\bfv|^2\dxt\leq 2^{d+2}.
\end{align*}
Combining the both cases yields (\ref{eq:3.34}) which implies together with the continuity of the maximal function on $L^2$,  (\ref{eq:cacc2}) and (\ref{eq:3.29})
\begin{align*}
\mathcal L^{d+1}\big(Q_1\cap \set{\mathcal M (|\nabla^2\bfv|^2)>N_1^2}\big)&\leq \mathcal L^{d+1}\big(Q_1\cap \set{\mathcal M (\chi_{Q_3}|\nabla^2\bfv-\nabla^2\bfh|^2)>N_0^2}\big)\\
&\leq\,\frac{c}{N_0^2}\int_{Q_3}|\nabla^2\bfv-\nabla^2\bfh|^2\dxt\\
&\leq\,\frac{c}{N_0^2}\int_{Q_3}|\bff|^2\dxt
\leq \frac{c}{N_0^2}\delta^2\\
&= \varepsilon\mathcal L^{d+1}(Q_1),
\end{align*}
choosing $\delta:=c^{-1/2}N_0\sqrt{\varepsilon}$.
So we have shown (\ref{imp2}) which yields (\ref{imp1}) by a scaling argument.\\
If (\ref{imp1})$_1$ holds then we have
\begin{align*}
\mathcal L^{d+1}&\big(Q_r\cap \set{\mathcal M (|\nabla^2\bfv|)^2>N_1^2}\big)\\&\leq \,\varepsilon\mathcal L^{d+1}\Big(Q_r\cap\set{\mathcal M(|\nabla^2\bfv|^2)>1}\cup \set{\mathcal M(|\bff|^2)>\delta^2}\Big)\\
&\leq \,\varepsilon\Big(\mathcal L^{d+1}\big(Q_r\cap\set{\mathcal M(|\nabla^2\bfv|^2)>1}\big)+\mathcal L^{d+1}\big(Q_r\cap \set{\mathcal M(|\bff|^2)>\delta^2}\big)\Big).
\end{align*}
Multiplying the equation for $\bfv$ by some small number $\varrho=\varrho(\|\bff\|_{q},\|\nabla^2\bfv\|_{2})$ we can assume that
\begin{align}
\mathcal L^{d+1}\big(Q_r\cap\set{\mathcal M(|\nabla^2\bfv|^2)>N_1^2}\big)<\varepsilon 
\end{align}
By induction we can establish that
\begin{align*}
\mathcal L^{d+1}&\big(Q_r\cap \set{\mathcal M (|\nabla^2\bfv|)^2>N_1^{2k}}\big)\\
&\leq \,{\varepsilon}^k\mathcal L^{d+1}\big(Q_r\cap\set{\mathcal M(|\nabla^2\bfv|^2)>1}+c\,\sum_{i=1}^k{\varepsilon}^i\mathcal L^{d+1}\big(Q_r \cap\set{\mathcal M(|\bff|^2)>\delta^2 N_1^{2(k-i)}}\big).
\end{align*}
In the induction step one has to introduce $\bfu_1:=\frac{\bfu}{N_1}$ which is a solution to the $\mathcal A$-Sokes problem with right hand side $\bff_1:=\frac{\bff}{N_1}$.
Since $\bff\in L^q(Q_0)$ and $q>2$ we have  $\mathcal M(|\bff|^2)\in L^{q/2}(Q_0)$ and hence
\begin{align*}
\int_{Q_r}&|\nabla^2\bfv|^q\dxt\leq\,\int_{Q_r}|\mathcal M(|\nabla^2\bfv|^2)|^{\frac{q}{2}}\dxt\\
&\leq  \,c\,\sum_{k=1}^\infty (N_1^q)^{2k}\mathcal L^{d+1}\big(Q_r\cap\set{\mathcal M(|\nabla^2\bfv|^2)>N_1^{2k}}\big)\\
&\leq \,c\,\sum_{k=1}^\infty (N_1^q)^{2k}\varepsilon^k\mathcal L^{d+1}\big(Q_r\cap\set{\mathcal M(|\nabla^2\bfv|^2)>1}\big)\\
&+\,c\,\sum_{k=1}^\infty (N_1^q)^{2k}\sum_{i=1}^k\varepsilon^i\mathcal L^{d+1}\big( Q_r\cap\set{\mathcal M(|\bff|^2)>\delta^2 N_1^{q(k-i)}}\big)\\
&\leq \,c\,\sum_{k=1}^\infty (\varepsilon N_1^q)^{2k}\Big(1+\int_{Q_r} |\mathcal M(|\bff|^2)|^{q/2}\dxt\Big).
\end{align*}
If we choose $\varepsilon N_1^q<1$ the sum is converging and we have $\nabla^2\bfv\in L^q(Q_r)$. Since the mapping $\bff\mapsto\nabla^2\bfv$ is linear we gain the desired estimate
\begin{align}\label{eq:Lqint}
\int_{Q_r}&|\nabla^2\bfv|^q\dxt\leq\,c\,\int_{Q_0}|\bff|^q\dxt.
\end{align}
\item[ii)] Now let $Q_1$ be a cube such that $4Q_1\cap (-\infty,0]\times\R^{d}\neq\emptyset$. Moreover, assume that $Q_1\cap Q_0\neq\emptyset$. We consider the solution $\tilde{\bfh}$ to 
\begin{align}
  \left\{\begin{array}{rlc}
     \partial_t\tilde{\bfh}- \Div\mathcal A(\ep(\tilde{\bfh})) &=\nabla \pi_{\tilde{\bfh}}& \mbox{in $\tilde{Q}_4$,}\\
      \Div \tilde{\bfh}&=0\qquad& \mbox{in $\tilde{Q}_4$,}\\
      \tilde{\bfh}&=\bfv\qquad\quad& \mbox{ \,on
        $\tilde{I}_4\times\partial B_4$,}\\
\tilde{\bfh}(t_0^4,\cdot)&=0\qquad\quad& \mbox{ \,in
        $B_4$,}
\end{array}\right.\label{eq:h}
\end{align}
where $\tilde{I}_m:=I_m\cap(0,T)$ and $\tilde{Q}_m:=\tilde{I}_m\times B_m$. We can establish a variant of (\ref{eq:cacc2}) on $\tilde{Q}_4$. Now we have $\sup_{\tilde{Q}_3}|\nabla^2\tilde{\bfh}|^2<\infty$ due to the smooth initial datum of $\tilde{\bfh}$ (recall that $\bfv(0,\cdot)=0$ a.e.). So we can finish the proof as before and gain  $\nabla^2\bfv\in L^q(Q_1)$. This implies again (\ref{eq:Lqint}).
\item[iii)] The situation $4Q_1\cap [T,\infty)\times\R^{d}\neq\emptyset$ is uncritical again and we can assume that ii) and iii) do not occur for the same cube (by choosing sufficiently small cubes)
\end{itemize}
Covering the set $(0,T)\times B$ by smaller cubes and combing i)-iii) yield the desired estimate.
\end{proof}



\begin{corollary}
\label{cor:Lqlocal}
Under the assumptions of Theorem \ref{thm:Lqlocal} we have for all balls $B\subset\R^d$ the following estimates.
\begin{itemize}
\item[a)] There is $c$ independent of $T$ such that
\begin{align*}
\int_0^T\int_{B}\bigg(\Big|\frac{\bfv}{T}\Big|^q+\Big|\frac{\nabla \bfv}{\sqrt{T}}\Big|^q+|\nabla^2 \bfv|^q\bigg)\dxt\leq \,c\,\int_{Q_0}|\bff|^q\dxt.
\end{align*}
\item[b)] We have $\partial_t\bfv\in L^q(0,T;L^q_{loc}(\R^d))$ together with 
\begin{align*}
\int_0^T\int_{B}|\partial_t\bfv|^q\dxt\,\leq \,c\,\int_{Q_0}|\bff|^q\dxt,
\end{align*}
where $c$ does not depend on $T$.
\item[c)] There is $\pi\in L^q((0,T),W^{1,q}_{loc}(\R^d))$ such that
\begin{align*}
\int_{Q_0} \bfv\cdot\partial_t\bfphi\dxt&-\int_{Q_0} \mathcal A(\ep(\bfv),\ep(\bfphi))\dx\\&=\int_{Q_0}\pi\,\Div\bfphi\dxt+\int_{Q_0}\bff\cdot\bfphi\dxt
\end{align*}
for all $\bfphi\in C^{\infty}_{0}([0,T)\times\R^{d})$.
\item[d)] There is $c$ independent of $T$ such that
\begin{align*}
\int_0^T\int_{B}\bigg(\Big|\frac{\pi}{\sqrt{T}}\Big|^q+|\nabla\pi|^q\bigg)\dxt\leq \,c\,\int_{Q_0}|\bff|^q\dxt.
\end{align*}
\end{itemize}
\end{corollary}
\begin{proof}
The estimate in a) is a simple scaling argument. Having a solution $\bfv$ defined on $(0,T)\times \R^d$ we gain a solution $\widehat{\bfv}$ on $(0,1)\times\R^d$ by setting
\begin{align*}
\widehat{\bfv}(s,x):=\frac{1}{T}\bfv(Ts,\sqrt{T}x).
\end{align*}
Now we apply Theorem \ref{thm:Lqlocal} to $\widehat{\bfv}$. The constant which appears is independent of $T$. Transforming back to $\bfv$ yields the claimed inequality.\\
b) For $\varphi\in C^\infty_0(Q_0)$ with $\bfphi(t,x)=\tau(t)\bfpsi(x)$ where $\bfvarphi\in C^\infty_0(G)$ ($G\in\R^d$ a bounded Lipschitz domain) we have
\begin{align*}
\int_{Q_0} \bfv\cdot\partial_t\bfphi\dxt&=\int_0^T\partial_t\tau\int_{\R^d} \bfv\cdot\big(\bfpsi_{\Div}+\nabla\bfPsi\big)\dxt\\
&=\int_0^T\partial_t\tau\int_{\R^d} \bfv\cdot\bfpsi_{\Div}\dxt\\
&=\int_{Q_0} \bfv\cdot\partial_t\bfphi_{\Div}\dxt
\end{align*}
where $\bfphi_{\Div}:=\bfphi-\nabla\Delta^{-1}_G\Div\bfphi$ and $\bfPsi:=\Delta^{-1}_G\Div\bfphi$.\footnote{Here $\Delta_G^{-1}$ is the solution operator to the Laplace equation with zero boundary data on $\partial G$.} Here we took into account $\Psi|_{\partial G}=0$ as well as $\Div\bfv=0$. Using $\nabla^2\bfv\in L^2(0,T;L^2_{loc}(\R^d))$ we proceed by%\marginpar{regularity $\Rightarrow$ no zero boundary values needed for test}
\begin{align*}
\int_{Q_0} \bfv\cdot\partial_t\bfphi\dxt
&=\int_0^T\int_{G} \big(\bff-\Div\mathcal A \ep(\bfv)\big)\cdot\bfphi_{\Div}\dxt\\
&\leq\,c\,\bigg(\int_0^T\int_G \big(|\nabla^2 \bfv|^q+\bff|^q\big)\dxt\bigg)^{\frac{1}{q}}\bigg(\int_0^T\int_G |\bfphi_{\Div}|^{q'}\dxt\bigg)^{\frac{1}{q'}}\\
&\leq\,c\,\bigg(\int_{Q_0} |\bff|^q\dxt\bigg)^{\frac{1}{q}}\bigg(\int_0^T\int_G |\bfphi|^{q'}\dxt\bigg)^{\frac{1}{q'}}.
\end{align*}
In the last step we used the estimate from Theorem \ref{thm:Lqlocal} and continuity of $\nabla\Delta_G^{-1}\Div$ on $L^{q'}(G)$. Duality implies $\partial_t\bfv\in L^q(0,T;L^q_{loc}(\R^d))$ and we can introduce the pressure function $\pi\in L^q(0,T;L^q_{loc}(\R^d))$ as claimed in b) by De Rahm's Theorem. Using the equation for $\bfv$ and the estimates in a) and b) we gain
\begin{align*}
\int_{Q}|\nabla\pi|^q\dxt\,\leq \,c\,\int_{Q_0}|\bff|^q\dxt.
\end{align*}
The estimate for $\pi$ in d) follows again by scaling.
\end{proof}

\begin{corollary}
\label{cor:Lqlocal+}
Let $\bff\in L^q(Q_0^+)$ for some $q>2$ where $Q_0^+:=(0,T)\times\R^d_+$, $\R^d_+=\R^d\cap[x_d>0]$, and let $\bfv\in L^1((0,T);\LD_{\Div}(\R^d_+))$ with $\bfv|_{x_d=0}=0$ be a weak solution to
\begin{align}
\int_{Q_0^+} \bfv\cdot\partial_t\bfphi\dxt-\int_{Q_0^+} \mathcal A(\ep(\bfv),\ep(\bfphi))\dx=\int_{Q_0^+}\bff\cdot\bfphi\dxt
\end{align}
for all $\bfphi\in C^{\infty}_{0,\Div}([0,T)\times \R^d_+)$.
Then the results from Theorem \ref{thm:Lqlocal} and Corollary \ref{cor:Lqlocal} hold for $\bfv$ for all half balls $B^+(z)\subset\R^d$ with $z_d=0$.\footnote{$B^+(z):=B(z)\cap[x_d>0]$}
\end{corollary}
\begin{proof}
We will show a variant of the $L^q$-estimate from Theorem \ref{thm:Lqlocal} on half balls $B^+$, i.e.
\begin{align}\label{eq:B+}
\int_0^T\int_{B^+}|\nabla^2\bfv|^q\dxt\leq \,c\,\int_0^T\int_{4B^+}|\bff|^q\dxt.
\end{align}
From this we can follow estimates in the fashion of Corollary \ref{cor:Lqlocal} as done there. In order to establish (\ref{eq:B+}) we will proceed as in the proof of
 Theorem \ref{thm:Lqlocal} replacing all balls with half ball. So let $Q_1\subset\R^{d+1}$ such that $4Q_1\subset Q_0$ (the other situation can be shown along the modifications indicated at the end of the proof of Theorem \ref{thm:Lqlocal}). Moreover, assume that $Q_1=I_1\times B_1(z)$ where $z_d=0$. We compare $\bfv$ with the unique solution $\bfh^+$ to
\begin{align}
  \left\{\begin{array}{rlc}
     \partial_t\bfh^+- \Div\mathcal A(\ep(\bfh^+)) &=\nabla \pi_{\bfh^+}& \mbox{in $Q_4^+$,}\\
      \Div \bfh^+&=0\qquad& \mbox{in $Q_4^+$,}\\
      \bfh^+&=\bfv\qquad\quad& \mbox{ \,on
        $I_4\times\partial B_4^+$,}\\
\bfh^+(0,\cdot)&=\bfv(0,\cdot)\qquad\quad& \mbox{ \,in
        $B_4^+$.}
\end{array}\right.\label{eq:h+}
\end{align}
We gain a version of the estimate (\ref{eq:cacc1}) on half-balls. In fact, there holds
\begin{align}\label{eq:cacc1+}
\sup_{t\in I_4}\int_{B_4^+}|\bfv-\bfh^+|^2\dx+\int_{Q_4^+}|\nabla\bfv-\nabla\bfh^+|^2\dxt\leq \,c\,\int_{Q_4^+}|\bff|^2\dxt.
\end{align}
Unfortunately, we cannot insert $\curl(\bfv-\bfh)$ as $\partial_d(\bfv-\bfh)$ is not necessarily zero on $[x_d=0]$. So we need some more subtle arguments. First we insert $\partial_t(\bfv-\bfh)$ which yields
\begin{align}\label{eq:cacc1+t}
\int_{Q_4^+}|\partial_t(\bfv-\bfh^+)|^2\dx+\sup_{t\in I_4}\int_{B_4^+}|\nabla\bfv-\nabla\bfh^+|^2\dx\leq \,c\,\int_{Q_4^+}|\bff|^2\dxt.
\end{align}
We can introduce the pressure terms $\pi_\bfv,\pi_{\bfh^+}\in L^q(I_4,L^q_0(B_4^+))$ in the equations for $\bfv$ and $\bfh^+$ and show
\begin{align}\label{eq:cacc1+pi}
\int_{Q_4^+}|\pi_\bfv-\pi_{\bfh^+}|^2\dx\leq \,c\,\int_{Q_4^+}|\bff|^2\dxt.
\end{align}
Estimate (\ref{eq:cacc1+pi}) is a consequence of the continuity of $\Bog:L^{2}_0(B_4^+)\rightarrow W^{1,2}_0(B_4^+)$ (see \cite{Bog}) and (\ref{eq:cacc1+t}): for $\varphi\in C^\infty_0(Q_4^+)$ there holds
\begin{align*}
\int_{Q_4^+}&(\pi_\bfv-\pi_{\bfh^+})\varphi\dxt=\int_{Q_4^+}(\pi_\bfv-\pi_{\bfh^+})\Div \Bog(\varphi-\varphi_{B_4^+})\dxt\\
&=\int_{Q_4^+}\mathcal A\big(\ep(\bfv-\bfh^+),\ep\big( \Bog(\varphi-\varphi_{B_4^+})\big)\big)\dxt+\int_{Q_4^+}\bff\cdot \Bog(\varphi-\varphi_{B_4^+})\big)\dxt\\
&-\int_{Q_4^+}\partial_t(\bfv-\bfh^+)\cdot \Bog(\varphi-\varphi_{B_4^+})\dxt\\
&\leq \,c\,\Big(\|\nabla(\bfv-\bfh^+) \|_2+\|\bff\|_2+\|\partial_t(\bfv-\bfh^+)\|_2\Big)\big\|\nabla\Bog(\varphi-(\varphi)_{B_4^+})\big\|_{2}\\
&\leq \,c\,\bigg(\int_{Q_4^+}|\bff|^2\dxt\bigg)^{\frac{1}{2}}\bigg(\int_{Q_4^+}|\varphi|^2\dxt\bigg)^{\frac{1}{2}}.
\end{align*}
Now we insert $\partial_\gamma(\eta^2\partial_\gamma(\bfv-\bfh))$ for $\gamma\in\set{1,...,d-1}$, where $\eta\in C^\infty_0(B_4)$ with $0\leq\eta\leq1$ and $\eta\equiv1$ on $B_3$. This yields together with (\ref{eq:cacc1+})-(\ref{eq:cacc1+pi})
\begin{align}\label{eq:cacc2+}
\int_{Q_3^+}|\tilde{\nabla}\nabla(\bfv-\bfh)|^2\dx\leq \,c\,\int_{Q_4^+}|\bff|^2\dxt,
\end{align}
where $\tilde{\nabla}:=(\partial_1,...,\partial_{d-1})$.
Finally, the only term which is missing is $\partial_d^2(\bfv-\bfh)$. On account of $\Div(\bfv-\bfh)=0$ we gain (cf. \cite{BKR})
\begin{align}\label{eq:dd}
|\partial_d^2(\bfv-\bfh)|&\leq \,c\,\big(|\tilde{\nabla}(\pi_\bfv-\pi_{\bfh^+})|+|\tilde{\nabla}\nabla(\bfv-\bfh)|+|\bff|\big).
\end{align}
So we have to estimate derivatives of the pressure. In fact we have
\begin{align}\label{eq:cacc1+dpi}
\int_{Q_3^+}|\tilde{\nabla}(\pi_\bfv-\pi_{\bfh^+})|^2\dx\leq \,c\,\int_{Q_4^+}|\bff|^2\dxt.
\end{align}
We can show this similarly to the proof of (\ref{eq:cacc1+pi}) replacing $\varphi$ by $\partial_\gamma\varphi$ and using (\ref{eq:cacc2+}). 
Combining (\ref{eq:cacc2+})- (\ref{eq:cacc1+dpi}) implies
\begin{align*}%\label{eq:cacc2+}
\int_{Q_3^+}|\nabla^2\bfv-\nabla^2\bfh|^2\dx\leq \,c\,\int_{Q_4^+}|\bff|^2\dxt.
\end{align*}
 Moreover, we know $\sup_{Q^+_3}|\nabla^2\bfh^+|^2<\infty$. Note that $\bfh^+=0$ on $Q_3\cap [x_d=0]$. This allows to show $\nabla^2\bfv\in L^q(Q^+_1)$ as in the proof of Theorem \ref{thm:Lqlocal}.  
\end{proof}

\begin{them}
\label{thm:Lqglobal}
Let  $Q:=(0,T)\times\Omega$ with a bounded domain $\Omega\subset\R^d$ having a $C^2$-boundary.
Let $\bff\in L^q(Q)$ for some $q>2$, where $q\in(1,\infty)$. Then there is a unique weak solution $\bfv\in L^\infty(0,T;L^2(B))\cap L^q(0,T;W^{1,q}_{0,\Div}(\Omega))$ to
\begin{align}
\int_{Q} \bfv\cdot\partial_t\bfphi\dxt-\int_{Q} \mathcal A(\ep(\bfv),\ep(\bfphi))\dx=\int_{Q}\bff\cdot\bfphi\dxt
\end{align}
for all $\bfphi\in C^{\infty}_{0,\Div}([0,T) \times \Omega)$ such that $\nabla^2\bfv\in L^q(Q)$. Moreover, we have
\begin{align*}
\int_{Q}|\nabla^2\bfv|^q\dxt\leq \,c\,\int_{Q}|\bff|^q\dxt.
\end{align*}
\end{them}
\begin{proof}
Due to the local $L^q$-theory for the whole space problem and the half-space problem which follow from Corollary \ref{cor:Lqlocal} and  Corollary \ref{cor:Lqlocal+}  (with the right scaling in $T$) the proof follows exactly as in \cite{So}, Thm. 4.1, in the case $\mathcal A=I$. Note that $L^q$-estimates for the stationary problem on bounded domains with given divergence are stated in Lemma \ref{lems:Aq}. 
\iffalse
We want to invert the operator
\begin{align*}
\mathscr L&:\mathscr Y\rightarrow L^q(I;L^q_{\Div}(B)),\\
\bfv&\mapsto\mathcal P_q\big(\partial_t\bfv-\Div\mathcal A\ep(\bfv)\big).
\end{align*}
Here $\mathcal P_q$ is the Helmholtz projection from $L^q(B)$ into $L^q_{\Div}(B)$. The latter one is defined by 
\begin{align*}
L^q_{\Div}(B):=\overline{C^\infty_{0,\Div}(B)}^{\|\cdot\|_q}.
\end{align*}
The Helmholtz-projection $\mathcal P_q \bfu$ of a function $\bfu\in L^q(B)$ can be defined as $\mathcal P_q \bfu:=\bfu-\nabla h$, where $h$ is the solution to the Neumann-problem
\begin{align*}
\begin{cases}\Delta h=\Div \bfu\quad\text{on}\quad B,\\
\mathcal N_B\cdot(\nabla h-\bfu)=0\quad\text{on}\quad\partial B.
\end{cases}
\end{align*}
The space $\mathscr Y$ is given by
\begin{align*}
\mathscr Y:= L^q(I;W^{1,q}_{0,\Div}\cap W^{2,q}(B))\cap W^{1,q}(I;L^q(B))\cap\set{\bfv(0,\cdot)=0,\,\,\bfv_{\partial B}=0}.
\end{align*}
Following \cite{So} (section 4) we will try to find an operator $\mathcal R:L^q(I;L^q_{\Div}(B))\rightarrow\mathscr Y$ such that
\begin{align}\label{eq:IS}
\mathscr L\circ\mathscr R=I+\tau
\end{align}
with $\|\tau\|<1$. The range of $\mathscr L\circ\mathscr R$ which then is $\bff\in L^q(0,T;L^q_{\Div}(B))$ is contained in the range of $\mathscr L$. So $\mathscr L$ is onto.\\
Let $\Omega_k$, $0=1,...,N$ be a covering of $\Omega$ such that $\Omega_0\Subset\Omega$ and $\Omega_k$ covers a (small) boundary strip of $\Omega$. There are local coordinates
\begin{align*}
z=\mathscr F_k y^k=(y_i^k,y_2^k,y_3^k-F_k(y_1^k,y_2^k)),\quad y^k=(y_1^k,y_2^k)\in B_\lambda(0)
\end{align*}
where $F_k$ is a $C^2$-function and $0<\lambda\leq\lambda_0<1$. After a rotation via a orthogonal matrix $V_k$ we end up with
$$z=\mathscr F_k V_k(x-\xi_k)=Z_k x,$$
where $\xi_k$ is the center point of $S_k=\partial\Omega\cap \Omega_k$. In this coordinate system we have a flat boundary which is contained in $\set{x_d=0}$. We consider a decomposition of unity $(\zeta_k)_{k=0}^N\subset C^\infty_0(\R^d)$ with respect to $\Omega_k$ such that $\mathrm{spt}\,\zeta_k\subset \Omega_k$.\\
We introduce for a function $u$ the notations
\begin{align*}
\mathscr Z_k u=u\circ Z_k^{-1},\quad \mathscr Z_k^{-1} u=u\circ Z_k,\quad\mathscr T_k u=(u_1,u_2,u_3-\partial_1 F_k \circ u_1-\partial_2 F_k \circ u_2).
\end{align*}
Furthermore, $\mathscr M_0 f$ is the extension of a function $f$ (by zero) to the whole space, $\sigma_k$ is the restriction of $f$ on $Q_k^T=(0,T)\times \Omega_k$ and $\sigma$ on $(0,T)\times B_\lambda^+(0)$. Note that if $\bff\in L^q_{\Div}(\Omega)$, then $\mathscr M_0 \bff\in L^q_{\Div}(\R^d)$. Finally, we denote by $\mathscr U \bff$, where $\bff\in L^q(\R^d_+)$ and 
$\mathscr U_0 \bff$, where $\bff\in L^q(\R^d)$ the solutions on half space and whole space respectively (see Corollary \ref{cor:Lqlocal} and  Corollary \ref{cor:Lqlocal+}). By $\mathscr Q \bff$
we denote the pressure corresponding to $\mathscr U \bff$. Now we define the operators
\begin{align*}
\mathscr R_0\bff&:=\zeta_0 \mathscr U_0 \mathscr M_0\bff+\sum_{k=1}^N\zeta_k \mathscr Z_k^{-1}\sigma V_k^{-1} \mathscr U \mathscr M \mathscr T_k V_k\mathscr Z_k\sigma_k\bff,\\
\mathscr P_0\bff&:=\sum_{k=1}^N\zeta_k \mathscr Z_k^{-1}\sigma V_k^{-1} \mathscr Q \mathscr M \mathscr T_k V_k\mathscr Z_k\sigma_k\bff.
\end{align*}
The idea is in the interior to extend the force to the whole space, compute the whole space solution and localize again. At the boundary it is more tricky since we have to flatten the problem before considering the half space problem (and of course we have to transform back after solving it).
Due to the involved cut-off functions $\mathscr R_0\bff$ is not necessary divergence-free. This will be corrected in the following way: we set
\begin{align*}
\mathscr R\bff=\mathscr R_0\bff+\mathscr R_1\bff,\quad \mathscr P\bff=\mathscr P_0\bff+\mathscr P_1\bff,
\end{align*}
where $\mathscr R_1\bff=\bfw$ and $\mathscr P_1\bff=s$ is the unique solution to the stationary problem
\begin{align*}
- \Div\mathcal A\ep(\bfw)+\nabla s=0,\quad \Div\bfw=\Div \mathscr R_0\bff,\quad \bfw|_{\partial\Omega}=0,
\end{align*}
cf. Lemma \ref{lems:Aq}. Now we clearly have $\mathscr R\bff\in \mathscr Y$. We need to establish (\ref{eq:IS}). It holds for $\tilde{\nabla}_k:=\nabla+\mathscr Z_k^{-1}V_k^{-1}\nabla F\frac{\partial}{\partial z_3}$
\begin{align}
\partial_t\mathscr R\bff&-\Div \mathcal A\ep(\mathscr R\bff)+\nabla \mathscr P\bff=\bff+\mathscr S+\partial_t\mathscr R_1\bff,\label{eq:4.10}\\
\mathscr S\bff&=-2\nabla \mathscr U_0 \mathscr M_0\bff\nabla\zeta_0-\Delta\zeta_0\mathscr U_0 \mathscr M_0\bff
-2\sum_{k=1}^N\nabla\bfu_k \nabla\zeta_k\nonumber\\&-\sum_{k=1}^N\Delta\zeta_k\bfu_k+\sum_{k=1}^N\nabla\zeta_k q^k
+\sum_{k=1}^N\zeta_k(\tilde{\nabla}^2_k-\nabla^2)\bfu^k\label{eq:4.11}\\
&-\sum_{k=1}^N\zeta_k(\tilde{\nabla}_k-\nabla)q^k
+\sum_{k=1}^N\zeta_k\mathscr Z_k^{-1}V_k^{-1}(\mathscr T_k-I)V_k\mathscr Z_k\sigma_k \bff\nonumber.
\end{align}
From (\ref{eq:4.10}) it follows
\begin{align*}
\mathscr L\mathscr R\bff=\bff+\mathcal P_q\mathscr S\bff+\mathcal P_q\partial_t\mathscr R_1\bff
\end{align*}
i.e. (\ref{eq:IS}) with $\tau=\mathcal P_q\mathscr S+\mathcal P_q\partial_t\mathscr R_1$. We need to estimate the norms of the operators $\mathscr S$ and $\partial_t\mathscr R_1$. If w echoose $T$ small enough the first to terms in (\ref{eq:4.11}) are small in accordance to Corollary \ref{cor:Lqlocal+}). The same is true for teh first three sums as a consequence of Corollary \ref{cor:Lqlocal}). We can argue similarly for the next three sums assuming that the gradients of the $F_k$ are small.\fi
\end{proof}


In order to treat problems with right hand side in divergence form
we consider the $\mathcal A$-Stokes operator 
$$\mathscr A_q:=-\mathcal P_q\Div\mathcal A\big(\ep(\cdot)\big),$$
where $\mathcal P_q$ is the Helmholtz projection from $L^q(B)$ into $L^q_{\Div}(B)$. The latter one is defined by 
\begin{align*}
L^q_{\Div}(B):=\overline{C^\infty_{0,\Div}(B)}^{\|\cdot\|_q}.
\end{align*}
The Helmholtz-projection $\mathcal P_q \bfu$ of a function $\bfu\in L^q(B)$ can be defined as $\mathcal P_q \bfu:=\bfu-\nabla h$, where $h$ is the solution to the Neumann-problem
\begin{align*}
\begin{cases}\Delta h=\Div \bfu\quad\text{on}\quad B,\\
\mathcal N_B\cdot(\nabla h-\bfu)=0\quad\text{on}\quad\partial B.
\end{cases}
\end{align*}
The $\mathcal A$-Stokes operator $\mathscr A_q$ enjoys the same properties than the Stokes operator $\mathbb A_q$ (see for instance \cite{GaSiSo}). 

For the $\mathcal A$-Stokes operator it holds $\mathcal D (\mathscr A_q)=W^{1,q}_{0,\Div}\cap W^{2,q}(\Omega)$ and
\begin{align}\label{eq:socont1}
&\|\bfu\|_{2,q}\leq c_1 \|\mathscr A_q\bfu\|_{q}\leq c_2\|\bfu\|_{2,q},\quad \bfu\in\mathcal D(\mathscr A_q),\\
&\int_\Omega \mathscr A_q \bfu\cdot\bfw\dx=\int_\Omega \bfu\cdot \mathscr A_{q'} \bfw\dx\quad\bfu\in D(\mathscr A_q),\,\,\bfw\in D(\mathscr A_{q'}).\label{eq:socont2}
\end{align}
Inequality (\ref{eq:socont1}) is a consequence of Lemma \ref{lems:Aq} a) and the continuity of $\mathcal P_q$.\\
Since $\mathscr A_q$ is positive its root $\mathscr A_q^{\frac{1}{2}}$ is well-defined with $\mathcal D (\mathscr A_q^{\frac{1}{2}})=W^{1,q}_{0,\Div}(\Omega)$ and
\begin{align}\label{eq:sorootcont1}
&\|\bfu\|_{1,q}\leq c_1\|\mathscr A^{\frac{1}{2}}_q\bfu\|_{q}\leq c_2\|\bfu\|_{1,q},\quad \bfu\in\mathcal D(\mathscr A^{\frac{1}{2}}_q),\\
&\int_\Omega \mathscr A^{\frac{1}{2}}_q \bfu\cdot\bfw\dx=\int_\Omega \bfu\cdot \mathscr A^{\frac{1}{2}}_{q'} \bfw\dx\quad\bfu\in D(\mathscr A^{\frac{1}{2}}_q),\,\,\bfw\in D(\mathscr A_{q'}^{\frac{1}{2}}).\label{eq:sorootcont2}
\end{align}
Finally, the inverse operator $\mathscr A_q^{-\frac{1}{2}}:L^q_{\Div}(\Omega)\rightarrow W^{1,q}_{0,\Div}(\Omega)$ is defined and it holds
\begin{align}\label{eq:ainv1}
&\|\nabla\mathscr A^{-\frac{1}{2}}_q\bfu\|_{q}\leq c\|\bfu\|_{q},\quad \bfu\in\mathcal D(\mathscr A^{-\frac{1}{2}}_q),\\
&\int_\Omega \mathscr A^{-\frac{1}{2}}_q \bfu\cdot\bfw\dx=\int_\Omega \bfu\cdot \mathscr A^{-\frac{1}{2}}_{q'} \bfw\dx\quad\bfu\in D(\mathscr A^{-\frac{1}{2}}_q),\,\,\bfw\in D(\mathscr A_{q'}^{-\frac{1}{2}}).\label{eq:ainv2}
\end{align}
From the definition of the square root of an positive self-adjoint operator follows also that
\begin{align*}
\mathscr A_{q'}^{\frac{1}{2}}&:W^{2,q'}\cap W^{1,q'}_{0,\Div}(\Omega)\rightarrow W^{1,q'}_{0,\Div}(\Omega),\\
\mathscr A_{q'}^{-\frac{1}{2}}&:W^{1,q'}_{0,\Div}(\Omega)\rightarrow W^{2,q'}\cap W^{1,q'}_{0,\Div}(\Omega),
\end{align*}
 together with
\begin{align}\label{eq:a1/2}
&\|\nabla\mathscr A^{\frac{1}{2}}_q\bfu\|_{q}\leq c\|\bfu\|_{2,q},\quad \bfu\in W^{2,q'}\cap W^{1,q'}_{0,\Div}(\Omega),\\
&\|\nabla\mathscr A^{-\frac{1}{2}}_q\bfu\|_{q}\leq c\|\bfu\|_{q},\quad \bfu\in W^{1,q'}_{0,\Div}(\Omega).\label{eq:a-1/2}
\end{align}

\iffalse
\begin{them}
  \label{lem:reg1}
Let $B\subset\R^d$ be a ball, $I=(t_0,t_1)$ be a bounded interval and $\bff\in L^q(Q)$ and $\bfw_0\in W^{2,q}\cap W^{1,q}_{0,\Div}(B))$, where $Q:=I\times B$ and $q\in(1,\infty)$. There is a unique solution $\bfw \in L^q(I;W^{2,q}\cap W^{1,q}_{0,\Div}(B))$ to
  \begin{align}
    \label{eq:vH'}
    \begin{aligned}
   \int_Q \bfw\cdot \partial_t\bfphi\dxt  -&\int_Q \mathcal{A}(\ep(\bfw),\ep(\bfphi)) \dxt\\
&   =\int_B \bfw_0\cdot\bfphi(t_0,\cdot)\dx+ \int_Q \bff\cdot\bfphi\dxt
    \end{aligned}
  \end{align}
  for all $\bfphi \in C^\infty_{0,\Div}([t_0,t_1)\times G)$. Moreover, we have
\begin{align*}
\dashint_{Q}|\nabla^2\bfw|^q\dxt\leq \,c\,\dashint_{Q}|\bff|^q\dxt+\,c\,\dashint_{B}|\nabla^2\bfw_0|^q\dx,
  \end{align*}
  where $c$ only depends on~$\mathcal{A}$ and $q$.
\end{them}
\begin{proof}
For simplicity let us assume $t_0=0$.
The operator $\mathscr A_q$ is a linear unbounded operator on $L^q(B)$
and the infinitesimal generator of a bounded analytic semigroup on $L^q(B)$.\marginpar{why??} We will denote this semigroup by $\mathcal S_p$. We will make use of the following property of analytic semigroups (see
\cite{Pa}[Chapter 2, Theorem 6.13]): for all $t>0$ the operator $\mathscr A_q\mathcal S_q(t)$ is bounded and
\begin{align}\label{eq:AS}
\|\mathscr A_q\mathcal S_q(t)\|_{\mathcal L(L^q(B),L^q(B))}\leq C(q)t^{-1}.
\end{align}
Moreover, the semigroup commutes with the operator $\mathscr A_q$.
We set
\begin{align*}
\bfu(t):=\mathcal S_p(t)\bfw_0+\int_{0}^t\mathcal S_q(t-\sigma)\bff(\sigma)\ds,\quad t\in I.
\end{align*}
which clearly is a weak solution to (\ref{eq:vH'}). We need $W^{2,q}$-estimates for the right hand side.
Let us consider the restriction of the operators $\mathcal S_q(t)$ to the space $W^{2,q}\cap W^{1,q}_{0,\Div}(B)$ and denote them by $\mathcal S_{2,q}(t)$. We obtain a strongly continuous semigroup generated by $\mathscr A_q$ (restricted to $W^{2,q}\cap W^{1,q}_{0,\Div}(B))$). 
We gain
\begin{align*}
\int_Q |\nabla^2\mathcal S_p(t)\bfw_0|^q\dxt&=\,\int_I\int_B |\nabla^2\mathcal S_{2,q}(t)\bfw_0|^q\dxt\\
&\leq\,c\,\int_I\int_B |\nabla^2\bfw_0|^q\dxt.
\end{align*}
For the integral involving $\bff$ we obtain using (\ref{eq:AS})
\begin{align*}
\int_B\Big|\mathscr A_q\int_{t_0}^t\mathcal S_q(t-\sigma)\bff(\sigma)\ds\Big|^q\dx&\leq \,c\,\int_{0}^t(t-\sigma)^{-q}\|\bff(\sigma)\|_{L^q(B)}^q\ds\\
&\leq \,c\,\int_\R\Big(\frac{\mathrm{sgn}(t-\sigma)}{|t-\sigma|}\chi_I(\sigma)\|\bff(\sigma)\|_{L^q(B)}\Big)^q\ds.
\end{align*}
The continuity of the Hilbert transform in $L^q(\R)$ and (\ref{eq:socont1}) yield
\begin{align*}
\int_Q\Big|\nabla^2\int_{0}^t\mathcal S_p(t-\sigma)\bff(\sigma)\ds\Big|^q\dxt&\leq \,c\,\int_Q\Big|\mathscr A_q\int_{t_0}^t\mathcal S_p(t-\sigma)\bff(\sigma)\ds\Big|^q\dxt \\
&\leq \,c\,\int_Q |\bff|^q\dxt
\end{align*}
which proves the claim.
\end{proof}
\fi 
Finally we state the main result of this section.
\begin{them}
  \label{lem:reg2}
Let  $Q:=(0,T)\times\Omega$ with a bounded domain $\Omega\subset\R^d$ having a $C^2$-boundary.
Let $\bfF\in L^q(Q)$, where $q\in(1,\infty)$. There is a unique solution $\bfw \in L^q(0,T;W^{1,q}_{0,\Div}(\Omega))$ to
  \begin{align}
    \label{eq:vH''}
    \begin{aligned}
   \int_Q \bfw\cdot \partial_t\bfphi\dxt  -&\int_Q \mathcal{A}(\ep(\bfw),\ep(\bfphi)) \dxt
   = \int_Q \bfF:\nabla\bfphi
    \end{aligned}
  \end{align}
  for all $\bfphi \in C^\infty_{0,\Div}([0,T)\times \Omega)$. Moreover we have
\begin{align*}
\dashint_{Q}|\nabla\bfw|^q\dxt\leq \,c\,\dashint_{Q}|\bfF|^q\dxt,
  \end{align*}
  where $c$ only depends on~$\mathcal{A}$ and $q$.
\end{them}
\begin{proof}
Let us first assume that $q>2$. Then Theorem \ref{thm:Lqglobal} applies.
We set $\bff:=\mathscr A_q^{-\frac{1}{2}}\Div\bfF$ which is defined via the duality
\begin{align*}
\int_\Omega \mathscr A^{-\frac{1}{2}}_q\Div\bfF \cdot\bfphi\dx=\int_\Omega \bfF :\nabla \mathscr A^{-\frac{1}{2}}_{q'}\bfphi\dx,\quad \bfphi\in C^\infty_{0,\Div}(\Omega),
\end{align*}
using (\ref{eq:ainv2}). So we gain $\bff\in L^q_{\Div}(\Omega)$ with 
\begin{align}\label{eq:fF}
\|\bff\|_q\leq \,c\,\|\bfF\|_q.
\end{align}
%Moreover, we consider $\tilde{\bfw}_0:=\mathscr A_q^{-\frac{1}{2}}\bfw_0$ and infer from (\ref{eq:a1/2})
%that $\tilde{\bfw}_0\in W^{2,q}\cap W^{1,q}_{0,\Div}(B)$ together with
%\begin{align}\label{eq:ww0}
%\int_B |\nabla^2\tilde{\bfw}_0|^q\dx\leq \,c\,\int_B |\nabla\bfw_0|^q\dx.
%\end{align}
We define $\tilde{\bfw}\in  L^q(0,T;W^{1,q}_{0,\Div}(\Omega))$ as the unique solution to
  \begin{align}
    \label{eq:vH'}
    \begin{aligned}
   \int_Q \tilde{\bfw}\cdot \partial_t\bfphi\dxt  -&\int_Q \mathcal A(\ep(\tilde{\bfw}),\ep(\bfphi)) \dxt=
   \int_Q \bff\cdot\bfphi\dxt
    \end{aligned}
  \end{align}
  for all $\bfphi \in C^\infty_{0,\Div}([0,T)\times \Omega)$. Theorem \ref{thm:Lqglobal} yields $\tilde{\bfw} \in L^q(0,T;W^{2,q}(\Omega))$ and
  \begin{align}\label{eq:wf}
\|\tilde{\bfw}\|_{2,q}\leq \,c\,\|\bff\|_q.
  \end{align}
  We want to return to the original problem and set $\bfw:=\mathscr A^{\frac{1}{2}}_q\tilde{\bfw}$ thus $\bfw\in L^q(0,T;W^{1,q}_{0,\Div}(\Omega))$. 
  Since $\mathscr A_{q'}^{\frac{1}{2}}:W^{1,q'}_{0,\Div}\cap W^{2,q'}(\Omega)\rightarrow W^{1,q'}_{0,\Div}(\Omega)$ we can replace
  $\bfphi$ by $\mathscr A_{q'}^{\frac{1}{2}}\bfphi$ in (\ref{eq:vH'}). This implies using (\ref{eq:sorootcont2}) and the definition of $\bff$
   \begin{align*}
   \int_Q \bfw\cdot \partial_t\bfphi\dxt  +\int_Q &\Div\mathcal A(\ep(\tilde{\bfw})):\mathscr A_{q'}^{\frac{1}{2}}\bfphi \dxt
   = \int_Q \bfF:\nabla\bfphi
  \end{align*}
   for all $\bfphi \in C^\infty_{0,\Div}(Q)$. On account of $\mathscr A_{q'}^{\frac{1}{2}}\bfphi\in W^{1,q'}_{0,\Div}(\Omega)$ and $\mathscr A_{q}^{\frac{1}{2}}\tilde{\bfw}\in W^{1,q}_{0,\Div}(\Omega)$ we gain due to (\ref{eq:sorootcont2})
   \begin{align*}
\int_Q \Div\mathcal A(\ep(\tilde{\bfw})):\mathscr A_{q'}^{\frac{1}{2}}\bfphi \dxt   &=\int_Q \mathscr A_{q}\tilde{\bfw}:\mathscr A_{q'}^{\frac{1}{2}}\bfphi \dxt 
 =\int_Q \mathscr A_{q}^{\frac{1}{2}}\tilde{\bfw}:\mathscr A_{q'}\bfphi \dxt\\
 &=\int_Q \bfw\cdot \Div\mathcal A(\ep(\bfphi)) \dxt=-\int_Q \mathcal A(\ep(\bfw),\ep(\bfphi)) \dxt
   \end{align*}
   using (\ref{eq:sorootcont2}) and $\bfw\in W^{1,q}_{0,\Div}(\Omega)$. This shows that $\bfw$ is the unique solution to (\ref{eq:vH}). Moreover,
   we obtain the desired regularity estimate via
   \begin{align*}
\int_{Q}|\nabla\bfw|^q\dxt&\leq \,c\,\int_{Q}|\mathscr A_q^{\frac{1}{2}}\bfw|^q\dxt=\,c\,\int_{Q}|\mathscr A_q\tilde{\bfw}|^q\dxt\\
&\leq\,c\,\int_{Q}|\nabla^2\tilde{\bfw}|^q
\leq \,\,c\,\int_{Q}|\bff|^q\dxt\\&\leq  \,c\,\int_{Q}|\bfF|^q\dxt
  \end{align*}
   as a consequence of (\ref{eq:sorootcont1}), the definition of $\bfw$, (\ref{eq:socont1}), (\ref{eq:wf}), and (\ref{eq:fF}). A simple scaling argument shows that
   the inequality is independent of the diameter of $I$ and $B$. So we have shown the claim for $q>2$.\\
   The case $q=2$ follows easily from a priori estimates and Korn's inequality. So let us
assume that $q<2$.  Duality arguments show that
  \begin{align*}
    \dashint_Q \abs{\nabla\bfw}^q\dxt = \, \sup_{\bfG \in L^{q'}(Q)}
    \bigg[ \dashint_Q \nabla\bfw:\bfG\dxt - \dashint_Q \abs{
      \bfG}^{q'}\dxt \bigg].
  \end{align*}
  For a given $\bfG\in L^{q'}(Q)$ let $\bfz_\bfG$ be the unique
  $L^{q'}(0,T;W^{1,q'}_{0,\Div}(\Omega))$-solution to
  \begin{align*}
  -\dashint_{Q} \bfz\cdot \partial_t\bfxi\dxt  + \int_Q \mathcal{A}(\ep(\bfz),\ep(\bfxi)) \dxt = \int_Q \bfG
    : \nabla \bfxi \dxt
  \end{align*}
  for all $\bfxi \in C^\infty_{0,\Div}([0,T)\times \Omega)$. Its existence together with the estimate
  \begin{align*}
    \dashint_Q|\nabla \bfz_{\bfG}|^{q'}\dxt\leq
    c\dashint_Q|\bfG|^{q'}\dxt
  \end{align*}
  follows from the first part of the proof as $q'>2$.
This, the density of
  $C^\infty_{0,\Div}(Q)$ and $\partial_t\bfz\in L^{q'}(0,T;W^{-1,q'}_{\Div}(\Omega))$ yield
  \begin{align*}
    \dashint_Q &\abs{\nabla\bfw}^q\dxt\\& \leq \, c\sup_{\bfG \in
      L^{q'}(Q)} \bigg[ \dashint_Q \mathcal{A}(\ep(\bfw),
    \ep(\bfz_\bfG))\dxt+\dashint_{Q} \partial_t\bfz_\bfG\cdot \bfu\dxt - \dashint_Q \abs{ \nabla\bfz_\bfG}^{q'}\dxt
    \bigg]
    \\
    &\leq c\, \sup_{\bfxi \in C^\infty_{0,\Div}(Q)} \bigg[ \dashint_Q
    \mathcal{A}(\ep(\bfw), \ep(\bfxi))\dxt-\dashint_{Q} \bfu\cdot \partial_t\bfxi\dxt- \dashint_Q \abs{\nabla
      \bfxi}^{q'}\dxt\bigg].
  \end{align*}
  The equation for $\bfw$ and Young's inequality imply
    \begin{align*}
    \dashint_Q \abs{\nabla\bfw}^q\dxt
    &\leq \,c\, \sup_{\bfxi \in C^\infty_{0,\Div}(Q)} \bigg[ \dashint_Q
    \bfF:\nabla\bfxi\dxt- \dashint_Q \abs{\nabla
      \bfxi}^{q'}\dxt\bigg]\\
      &\leq\,c\,\dashint_Q|\bfF|^{q}\dxt
  \end{align*}
  and hence the claim.
\end{proof}

\newpage

\section{Solenoidal Lipschitz truncation}
\label{sec:soltrunc}
The purpose of the Lipschitz truncation technique is to approximate a
Sobolev function~$u \in W^{1,p}$ by $\lambda$-Lipschitz
functions~$u_\lambda$ that coincides with~$u$ up to a set of small
measure. The functions $u_\lambda$ are constructed nonlinearly by
modifying~$u$ on the level set of the Hardy-Littlewood maximal
function of the gradient~$\nabla u$.  This idea goes back to Acerbi
and Fusco~\cite{AF}.  Lipschitz truncations are used in various
areas of analysis: calculus of variations, in the existence theory of
partial differential equations, and in the regularity theory. We refer
to~\cite{DMS} for a longer list of references.
The Lipschitz truncation in the context of parabolic PDE's can be found in \cite{KinLew02} and \cite{DRW}. The solenoidal Lipschitz truncation in the non-stationary setting is introduced in \cite{BrDS}. We present a version of it which is appropriate for our purposes. For notational simplicity we assume $d=3$ and refer to \cite{BrDS} (remark 2.1) for the higher dimensional case.
\begin{them}
  \label{cor:appl}
Let $\bfu\in L^\sigma(I;W^{1,\sigma}_{\div}(B))\cap L^\infty(I;L^\sigma(B))$ with $\partial_t\bfu=\Div\bfH$ in $D'_{\Div}(Q)$ with $\bfH\in L^\sigma(Q)$. Then for every $m_0\gg1$ and $\gamma>0$
there exist $\lambda\in[2^{m_0}\gamma,2^{2m_0}\gamma]$ and a function $\bfu_\lambda$ with the following properties
 \begin{enumerate}[label=(\alph{*})]
  \item \label{itm:1} It holds $\bfu_\lambda\in L^\infty(I, W^{1,\infty}_{0,\Div}(B))$ with $\|\nabla\bfu_\lambda\|_{\infty}\leq c\lambda$.
  \item \label{itm:2} We have
  \begin{align*}
   \lambda^{\sigma} \frac{\mathcal{L}^{d+1}\left(\frac 12 Q\cap\set{\bfu_\lambda \not=
          \bfu}\right)}{\abs{Q}} &\leq \frac{c}{m_0} \bigg(\dashint_{Q}
    r_B^{-\sigma}\abs{\bfu}^{\sigma}+\abs{\nabla \bfu}^{\sigma}\dxt+\dashint_{Q} \abs{ \bfH}^{\sigma}\dxt\bigg).
    \end{align*}
  \item \label{itm:3} It holds
\begin{align*}  
  \dashint_{Q}
    \abs{\bfu_\lambda}^{\sigma}\dxt&\leq c\bigg(\dashint_{Q}
    \abs{\bfu}^{\sigma}+\dashint_{Q} r_B^{\sigma}\abs{ \bfH}^{\sigma}\dxt\bigg),\\
  \dashint_{Q}\abs{\nabla \bfu_\lambda}^{\sigma}\dxt&\leq c\bigg(\dashint_{Q}
    r_B^{-\sigma}\abs{\bfu}^{\sigma}+\abs{\nabla \bfu}^{\sigma}\dxt+\dashint_{Q} \abs{ \bfH}^{\sigma}\dxt\bigg).
  \end{align*}
   \item \label{itm:4} We have $\partial_t(\bfu-\bfu_\lambda)\in L^{\sigma'}(\frac 12 I,W^{-1,\sigma'}(\frac 12 B))$ and
   \begin{align*}
   -\dashint_{\frac 12 Q} &(\bfu-\bfu_\lambda)\cdot\partial_t\bfphi\dxt\\
  & \leq c(\kappa)\int_{\frac 12 Q} \chi_{\mathcal O_\lambda}|\nabla\bfphi|^{\sigma'}\dxt+\kappa \bigg(\dashint_{Q}
    r_B^{-\sigma}\abs{\bfu}^{\sigma}+\abs{\nabla \bfu}^{\sigma}+ \abs{ \bfH}^{\sigma}\dxt\bigg)
   \end{align*}
   for all $\bfphi\in C^\infty_0(\frac 12 Q)$ and all $\kappa>0$.
  \end{enumerate}
\end{them}

\begin{proof}

Let $\gamma \in C_0^\infty(\Bz)$ with$ \chi_{\frac 12 \Bz} \leq \gamma
\leq \chi_{\Bz}$, where $\Bz$ is a ball. Let $A$ denote the annulus $\Bz \setminus \frac 12 \Bz$.
Then according to \cite{Bog} there exists a \Bogovskii{}
operator $\Bog_A$ from\footnote{$C^\infty_{0,0}$ is the subspace of
  $C^\infty_0$ whose elements have mean value zero.}
$C^\infty_{0,0}(A) \to C^\infty_0(A)$ which is bounded from $L^q_0(A)
\to W^{1,q}_0(A)$ for all $q \in (1,\infty)$ such that $\divergence
\Bog_A = \identity$.  Define
\begin{align*}
  \tilde{\bfu} &:= \gamma \bfu - \Bog_A( \divergence(\gamma \bfu)) =
  \gamma \bfu - \Bog_A( \nabla\gamma \cdot \bfu).
\end{align*}
Then $\divergence \tilde{\bfu}=0$ on $\Iz \times \Bz$ and
$\tilde{\bfu}(t) \in W^{1,\sigma}_0(\Bz)$, so we can extend
$\tilde{\bfu}$ by zero in space to $\tilde{\bfu} \in L^\sigma(I,
W^{1,\sigma}_{\divergence}(\Rdr))$.  Since $\tilde{\bfu}=\bfu$ on $I
\times \frac 12 \Bz$, we have
\begin{align}
  \label{eq:uH2}
  \int_{\Qz} \partial_t \tilde{\bfu} \cdot \bfxi \dxt &= \int_{\Qz}
  \bfH : \nabla \bfxi \dxt \qquad \text{for all } \bfxi \in
  C^\infty_{0,\divergence}(\tfrac 12\Qz).
\end{align}
Now, we define pointwise in time
\begin{align*}
  \bfw := \curl^{-1} (\tilde{\bfu}) = \curl^{-1}\big(   \gamma \bfu -
  \Bog_{\Bz \setminus \frac 12 \Bz}( \nabla\gamma \cdot \bfu) \big).
\end{align*}
It follows from Lemma 2.1  in \cite{BrDS}) and continuity properties of $\Bog$ that
\begin{align}
\label{eq:w}
\begin{aligned}
\int_{\frac{1}{2}Q}|\nabla\bfw|^{\sigma}\dxt&\leq\,c\,\dashint_{Q}
    \abs{\bfu}^{\sigma}\dxt,\\
\int_{\frac{1}{2}Q}|\nabla^2\bfw|^\sigma\dxt&\leq\,c\,\bigg(\,\,\dashint_{Q}r_B^{-\sigma}
    \abs{\bfu}^{\sigma}\dxt+\int_Q\abs{\nabla \bfu}^{\sigma}\dxt\bigg).
\end{aligned}
\end{align}
For a ball $B' \subset \Rdr$ and a function $f \in L^s(B')$ we define $\Delta^{-2}_{B'} \Delta f$ as the weak solution $F \in
W^{2,s}_0(B')$ of
\begin{align}
  \label{eq:defbi2}
  \int_{B'}\Delta F\Delta\phi\dx &= \int_{B'}f\Delta\phi\dx \qquad
  \text{for all }\phi \in C^\infty_0(B').
\end{align}
Then $f - \Delta (\Delta^{-2}_{B'} \Delta f)$ is harmonic on~$B'$. We define $\bfz(t) := \bfz_{\frac 12 \Qz}(t) = \Delta \Delta_{\frac 12
  \Bz}^{-2} \Delta \bfw(t)$ for $t \in \frac12 \Iz$, then
\begin{align}
  \label{eq:zt}
  \int_{\Qz} \bfz\cdot \partial _t \Delta\bfpsi \dxt &= \int_{\Qz}
  \bfw\cdot \partial _t \Delta\bfpsi \dxt = - \int_{\Qz} {\bfH}:
  \nabla^2\bfpsi \dxt,
\end{align}
for all $\bfpsi \in C^\infty_0(\tfrac 12 \Qz)$.  Since $\Delta_{\frac
  12 \Bz}^{-2} \bfw(t) \in W^{2,s}_0(\frac 12 \Bz)$, we can extend it
by zero to a function from $W^{2,s}(\Rdr)$. In this sense it is natural
to extend $\bfz(t)$ by zero to a function $L^s(\Rdr)$. As a consequence of \cite[Lemma 2.3]{BrDS} and (\ref{eq:w}) we have
\begin{align}
\label{eq:z0}
\begin{aligned}
\int_{\frac{1}{2}Q}|\nabla\bfz|^{\sigma}\dxt&\leq\,c\,\dashint_{Q}
    \abs{\bfu}^{\sigma}\dxt,\\
\int_{\frac{1}{2}Q}|\nabla^2\bfz|^{\sigma}\dxt&\leq\,c\,\bigg(\,\,\dashint_{Q}r_B^{-{\sigma}}
    \abs{\bfu}^{\sigma}\dxt+\int_Q\abs{\nabla \bfu}^{\sigma}\dxt\bigg),\\
\int_{\frac{1}{2}Q}|\partial_t\bfz|^{\sigma}\dxt&\leq\,c\,\int_Q\abs{\bfH}^{\sigma}\dxt
\end{aligned}
\end{align}


For $\lambda,\alpha>0$ and $\sigma>1$ we define
% \comment{L+S: Why don't we cut of $\partial_t\bfz$ directly?}
\begin{align}
  \label{eq:defOal}
  \Oal &:= \set{\Mas(\chi_{\frac 13 \Qz} \abs{\nabla^2
      \bfz})>\lambda} \cup \set{\alpha \Mas(\chi_{\frac 13 \Qz}
    \abs{\partial_t\bfz}) > \lambda}.
\end{align}
We decompose $\Oal$ into a family of parabolic Whitney cubes $(Q_i)_{i\in\N}$ an consider a decomposition of unity $(\varphi)_{i\in\N}$ with respect to it as done in \cite{BrDS} after (2.11).
We define
\begin{align*}
  \mathcal{I} := \set{ i \,:\, Q_i \cap \tfrac 14 \Qz \neq \emptyset}.
\end{align*}
Taking $\lambda$ large enough, the continuity of the maximal function (\ref{eq:Mascont}) implies $Q_i \subset \frac 13 \Qz$ (and
$Q_j \subset \frac 13 \Qz$ for $j \in A_i$) for all $i \in
\mathcal{I}$.  
  For each $i \in \mathcal{I}$ we define local approximation $\bfz_i$ for $\bfz$ on $Q_i$ by
  \begin{align}
    \label{eq:def_zj}
    \bfz_i &:=
    \Pi^0_{I_i}\Pi^1_{B_i}(\bfz),
  \end{align}
  where $\Pi^1_{B_j}(\bfz)$ is the first order averaged Taylor
  polynomial \cite{BreSco94,DieR07} with respect to space and $\Pi^0_{I_i}$ is the zero order
averaged Taylor polynomial in time. We set
\begin{align}
  \label{eq:defwlalt}
  \zal &=
  \begin{cases}
    \bfz & \qquad \text{on $\tfrac 14 \Qz \setminus \Oal$},
    \\
    \sum_{i \in \mathcal{I}} \phi_i \bfz_i &\qquad \text{on $\tfrac 14
      \Qz \cap \Oal$}.
  \end{cases}
\end{align}
We apply the arguments used in the proof of Theorem \cite{BrDS} (Thm. 2.2) to the constant sequence $\bfu$ with the choice $\alpha=1$. So we have
\begin{align}
    \label{eq:def-ulambda}
    \bfu_\lambda:=\curl(\zeta\bfz_\lambda)+\curl(\zeta(\bfw-\bfz)),
  \end{align}
  where $\zeta\in C^\infty_0(\frac 16 Q)$ with $\chi_{\frac 18
    Q}\leq\zeta\leq \chi_{\frac 16 Q}$. This means on $\frac 18
  Q$ it holds
  \begin{align*}
    \bfu_\lambda&= \bfu + \curl(\bfz_\lambda-\bfz).
  \end{align*}
We immediately obtain the claim of \ref{itm:1}. As a consequence of
the Lemmas 2.1, 2.4 and 2.9 in \cite{BrDS}
we gain the inequalities 
\begin{align}  \label{eq:z}
\begin{aligned}
  \dashint_{\frac{1}{4} Q}
    \abs{\nabla \bfz_\lambda}^{\sigma}\dxt&\leq c\bigg(\dashint_{Q}
    \abs{\bfu}^{\sigma}+\dashint_{\frac{1}{3}Q} r_B^{\sigma}\abs{ \partial_t\bfz}^{\sigma}\dxt\bigg),\\
  \dashint_{\frac{1}{4}Q}\abs{\nabla^2 \bfz_\lambda}^{\sigma}\dxt&\leq c\bigg(\dashint_{Q}r_B^{-\sigma}
    \abs{\bfu}^{\sigma}+\abs{\nabla \bfu}^{\sigma}\dxt+\dashint_{\frac 13 Q} \abs{ \partial_t\bfz}^{\sigma}\dxt\bigg),\\
  \dashint_{\frac{1}{4}Q}\abs{\partial_t \bfz_\lambda}^{\sigma}\dxt&\leq c\bigg(\dashint_{Q}r_B^{-\sigma}
    \abs{\bfu}^{\sigma}+\abs{\nabla \bfu}^{\sigma}\dxt+\dashint_{\frac 13 Q} \abs{ \partial_t\bfz}^{\sigma}\dxt\bigg),
\end{aligned}
  \end{align}
which imply the estimates
\begin{align*}  
  \dashint_{Q}
    \abs{\bfu_\lambda}^{\sigma}\dxt&\leq c\bigg(\dashint_{Q}
    \abs{\bfu}^{\sigma}+\dashint_{Q} r_B^{\sigma}\abs{ \partial_t\bfz}^{\sigma}\dxt\bigg),\\
  \dashint_{Q}\abs{\nabla \bfu_\lambda}^{\sigma}\dxt&\leq c\bigg(\dashint_{Q}r_B^{-\sigma}
    \abs{\bfu}^{\sigma}+\abs{\nabla \bfu}^{\sigma}\dxt+\dashint_{Q} \abs{ \partial_t\bfz}^{\sigma}\dxt\bigg).
  \end{align*}
Finally we can replace $\partial_t\bfz$ by $\bfH$ on account of (\ref{eq:z0}).\\
It remains to find good levels. So we set
  \begin{align*}
    g &:= \mathcal M_s(\chi_{\frac 13 Q}\abs{\nabla^2 \bfz})+  \mathcal M_s(\chi_{\frac 13 Q}\abs{\partial_t\bfz})
  \end{align*}
  and gain from the continuity of $\mathcal M_s$ and (\ref{eq:z0})
  \begin{align}\label{eq:solparaneu}
\begin{aligned}
  \int_{\R^{d+1}} \abs{g}^\sigma \,\dx &\leq c\bigg(\int_{\frac{1}{3} Q}
    \abs{\nabla^2 \bfz}^{\sigma}\dxt+\int_{\frac{1}{3} Q} \abs{ \partial_t\bfz}^{\sigma}\dxt\bigg)\\
&\leq c\bigg(\int_{Q}
    r_B^{-\sigma}\abs{\bfu}^{\sigma}+\abs{\nabla \bfu}^{\sigma}\dxt+\int_{Q} \abs{ \bfH}^{\sigma}\dxt\bigg).
\end{aligned}
  \end{align}
  Furthermore, it holds for every $m_0\in\N$ and every $\gamma>0$
    \begin{align*}
    \int_{\R^{d+1}} \abs{g}^\sigma \,\dx &= \int_{\R^{d+1}} \int_0^\infty
    \sigma t^{\sigma-1} \chi_{\{\abs{g}>t\}} \,\dt \,\dx\\& \geq
    \int_{\R^{d+1}} \sum_{m= m_0}^{2m_0-1} \sigma (2^m\gamma)^{\sigma}
    \chi_{\{\abs{g}>\gamma 2^{m+1}\}} \,\dx.
  \end{align*}
So, there is $m_1\in\set{m_0,...,2m_0-1}$ such that
  \begin{align*}
   \int_{\R^{d+1}} (2^{m_1}\gamma)^{\sigma}
    \chi_{\{\abs{g}>\gamma 2^{m_1+1}\}} \,\dx\leq \frac{c}{m_0} \int_{\R^{d+1}} \abs{g}^\sigma \,\dx .
  \end{align*}
  Setting $\lambda=\gamma 2^{m_1+1}$ we obtain
    \begin{align*}
   \lambda^{\sigma}
    |\tfrac{1}{3} Q\cap\set{\abs{g}>\lambda}|\leq \frac{c}{m_0} \int_{\R^{d+1}} \abs{g}^\sigma \,\dx .
  \end{align*}
  Combining this with (\ref{eq:solparaneu}) gives the estimate in b) due to the definition of $\mathcal O_\lambda$. \\ 
  We have $\bfu_\lambda-\bfu=\curl(\bfz_\lambda-\bfz)$ on $\frac{1}{8}Q$ such that (\ref{eq:z0}) and (\ref{eq:z}) imply
  $$\partial_t(\bfu_\lambda-\bfu) \in L^{\sigma'}\big(\tfrac 18 I,W^{-1,\sigma'}(\tfrac 18 B)\big)$$. Moreover, we gain for $\bfphi\in C^\infty_0(\frac{1}{8}Q)$
  \begin{align*}
   -\int_{\frac 18 Q} (\bfu-\bfu_\lambda)\cdot\partial_t\bfphi\dxt=\int_{\frac 18 Q} \chi_{\mathcal O_\lambda}\partial_t(\bfz-\bfz_\lambda)\cdot\curl\bfphi\dxt\\
    \leq c(\kappa)\int_{\frac 12 Q} \chi_{\mathcal O_\lambda}|\nabla\bfphi|^{\sigma'}\dxt+\kappa \bigg(\dashint_{Q}
    \abs{\partial_t(\bfz_\lambda-\bfz)}^{\sigma}\dxt\bigg)
   \end{align*}
   as a consequence of Young's inequality. Applying (\ref{eq:z0}) and (\ref{eq:z}) yields
\begin{align*}    
\dashint_{Q}
    \abs{\partial_t(\bfz_\lambda-\bfz)}^{\sigma}\dxt
&\leq \,c\,\bigg(\dashint_{Q}
    r_B^{-\sigma}\abs{\bfu}^{\sigma}+\abs{\nabla \bfu}^{\sigma}\dxt+\dashint_{Q} \abs{ \bfH}^{\sigma}\dxt\bigg).
 \end{align*}  
 So we have shown the estimate claimed in \ref{itm:4} on $\frac{1}{8}Q$.\\
  A simple scaling argument allows us to obtain the desired estimates on $\frac 12 Q$.
\end{proof}

\newpage

\section{$\mathcal{A}$-Stokes approximation - evolutionary case}
\label{sec:Astokespara}

For a function $\bfw\in L^1(Q)$ with $\partial_t \bfw \in L^{q'}(J;W^{-1,q'}_{\Div}(B))$ we introduce
the unique function $\bfH_\bfw\in L^{q'}_0(Q)$ with
\begin{align*}
\int_{Q} \bfw\cdot \partial_t\bfvarphi\dxt=\int_Q \bfH_\bfw:\nabla\bfvarphi\dxt
\end{align*}
for all $\bfvarphi\in C^\infty_{0,\Div}(Q)$.
We begin with a variational inequality for the non-stationary $\mathcal{A}$-Stokes
system.


\begin{lemma}
  \label{lem:varineqS}
  For all balls~$B\subset\R^d$, all bounded intervals $J=(t_t,t_1)$ and $\bfu\in C^0_w([0,T];L^1(B))\cap L^q(J;W^{1,q}(B))$ with $\bfu(0,\cdot)=0$ a.e. it holds for $Q=J\times B$
  \begin{align*}
%\begin{aligned}
    %\label{eq:Liptrunvi}
    \dashint_Q &\abs{\nabla\bfu}^q\dxt
    \leq \,c\, \sup_{\bfxi \in C^\infty_{0,\Div}(Q)} \bigg[ \dashint_Q\Big(
    \mathcal{A}(\ep(\bfu), \ep(\bfxi))- \bfu\cdot \partial_t\bfxi\Big)\dxt\\&\qquad\qquad\qquad\qquad\qquad\qquad - \dashint_Q \Big(\abs{\nabla
      \bfxi}^{q'}+|\bfH_\bfxi|^{q'}\Big)\dxt\bigg].
%\end{aligned}
  \end{align*}
  where $c$ only depends on~$\mathcal{A}$ and $d$.
\end{lemma}
\begin{proof}
  Duality arguments show that
  \begin{align*}
    \dashint_Q \abs{\nabla\bfu}^q\dxt = \, \sup_{\bfG \in L^{q'}(Q)}
    \bigg[ \dashint_Q \nabla\bfu:\bfG\dxt - \dashint_Q \abs{
      \bfG}^{q'}\dxt \bigg].
  \end{align*}
  For a given $\bfG\in L^{q'}(Q)$ let $\bfz_\bfG$ be the unique
  $L^{q'}(J;W^{1,q'}_{0,\Div})$-solution to
  \begin{align*}
  -\dashint_{Q} \bfz\cdot \partial_t\bfxi\dxt  + \int_Q \mathcal{A}(\ep(\bfz),\ep(\bfxi)) \dxt = \int_Q \bfG
    : \nabla \bfxi \dxt
  \end{align*}
  for all $\bfxi \in C^\infty_{0,\Div}([t_0,t_1)\times B)$. We also have that $\partial_t\bfz_\bfG\in L^{q'}(J;W^{-1,q'}_{\Div}(B))$.
  Due to Theorem \ref{lem:reg2} this solution
  satisfies 
  \begin{align*}
    \dashint_Q|\nabla \bfz_{\bfG}|^{q'}\dxt+\dashint_{Q}|\bfH_{\bfz_\bfG}|^{q'}\dxt\leq
    c\dashint_Q|\bfG|^{q'}\dxt.
  \end{align*}
  In other words, the mapping $L^{q'}(B)\ni\bfG\mapsto\bfz_\bfG\in
 L^{q'}(J; W^{1,q'}_{0,\Div}(B))$ is continuous.  This and the density of
  $C^\infty_{0,\Div}(Q)$ gives due to $\bfu(0,\cdot)=0$
  \begin{align*}
    \dashint_Q &\abs{\nabla\bfu}^q\dxt\\& \leq \, c\sup_{\bfG \in
      L^{q'}(Q)} \bigg[ \dashint_Q \mathcal{A}(\ep(\bfu),
    \ep(\bfz_\bfG))\dxt+\dashint_{Q} \partial_t\bfz_\bfG\cdot \bfu\dxt \\&\qquad\qquad\qquad\qquad\qquad\qquad- \dashint_Q \Big(\abs{ \nabla\bfz_\bfH}^{q'}+|\bfH_{\bfz_\bfG}|^{q'}\Big)\dxt
    \bigg]
    \\
    &\leq c\, \sup_{\bfxi \in C^\infty_{0,\Div}(Q)} \bigg[ \dashint_Q
    \mathcal{A}(\ep(\bfu), \ep(\bfxi))\dxt-\dashint_{Q} \bfu\cdot \partial_t\bfxi\dxt\\
    &\qquad\qquad\qquad\qquad\qquad\qquad - \dashint_Q \Big(\abs{\nabla
      \bfxi}^{q'}+|\bfH_\bfxi|^{q'}\Big)\dxt\bigg]
  \end{align*}
which yields the claim.
\end{proof} 
Let us now state the $\mathcal{A}$-Stokes approximation. In the following let $B$ be a ball with radius~$r$ and $J$ an interval with length $2r^2$. Let $\tilde{Q}$ denote either
  $Q=J\times B$ or $2Q$. We use similar notations for $\tilde{J}$ and $\tilde{B}$.
\begin{them}
  \label{Astokes}
 Let $\bfv\in L^{qs}(2\tilde{J}; W^{1,qs}_{\Div}(2\tilde{B}))$, $q,s>1$, be an
  {almost $\mathcal{A}$-Stokes solution}
 in the sense that
  \begin{align}
  \label{eq:almAStokespara}
  \begin{aligned}
    \bigg|\dashint_{2Q} \bfv\cdot \partial_t\bfxi\dxt&-\dashint_{2Q} \mathcal{A}(\ep(\bfv),\ep(\bfxi))\dxt\bigg|\leq
    \delta \dashint_{2\tilde{Q}} |\ep(\bfv)|\dxt\,\|\nabla\bfxi\|_\infty
    \end{aligned}
  \end{align}
  for all $\bfxi\in C^{\infty}_{0,\Div}(2Q)$ and some
  small~$\delta>0$. Then the unique solution $\bfw \in
  L^q(J;W^{1,q}_{0,\Div}(B))$ to
  \begin{align}
    \label{eq:vH}
    \begin{aligned}
   \int_Q \bfw\cdot \partial_t\bfxi\dxt  &-\int_Q \mathcal{A}(\ep(\bfw),\ep(\bfxi)) \dxt\\
   &= \int_Q \bfv\cdot \partial_t\bfxi\dxt -
    \int_Q \mathcal{A}(\ep(\bfv),\ep(\bfxi)) \dxt
    \end{aligned}
  \end{align}
  for all $\bfxi \in C^\infty_{0,\Div}([t_0,t_1)\times B)$ satisfies
  \begin{align*}
\dashint_{Q} \Big|\frac{\bfw}{r}\Big|^q\dxt+\dashint_{Q}
    |\nabla\bfw|^q\dxt\leq \kappa\bigg(\dashint_{2\tilde{Q}}
    |\nabla\bfv|^{qs}\dxt\bigg)^{\frac{1}{s}}.
  \end{align*}
  It holds $\kappa=\kappa(q,s,\delta)$ and
  $\lim_{\delta\rightarrow0}\kappa(q,s,\delta)=0$. The function $\bfh
  := \bfv - \bfw$ is called the $\mathcal{A}$-Stokes approximation of
  $\bfv$. 
\end{them}
\begin{remark}
From the proof of Theorem \ref{Astokes} we gain the following stability result choosing $p=q s=q$.
\begin{align*}
 \dashint_{Q} \Big|\frac{\bfw}{r}\Big|^p\dxt+\dashint_{Q}
    |\nabla\bfw|^p\dxt\leq c\,\dashint_{2\tilde{Q}}
    |\nabla\bfv|^{p}\dxt.
  \end{align*}
  Indeed $\kappa$ stays bounded if $s\rightarrow 1$.
\end{remark}

\begin{proof}
  Let $\bfw$ be defined as in~\eqref{eq:vH}.  Combining Poincar{\'e}'s inequality with Lemma \ref{lem:varineqS}
  and~~\eqref{eq:vH} shows
  \begin{align}
    \label{eq:Liptrunvib}
    \begin{aligned}
   \dashint_{Q} &\Big|\frac{\bfw}{r}\Big|^q\dxt+\dashint_{Q}
      |\nabla\bfw|^q\dxt\\& \leq c\, \sup_{\bfxi \in
        C^\infty_{0,\Div}(Q)} \bigg[ \dashint_Q \mathcal{A}(\ep(\bfv),
      \ep(\bfxi))\dxt-\dashint_Q \bfv\cdot \partial_t\bfxi\dxt \\
    &\qquad\qquad\qquad\qquad\qquad\qquad- \dashint_Q \Big(\abs{\nabla \bfxi}^{q'}+|\bfH_\bfxi|^{q'}\Big)\dxt\bigg].
      \end{aligned}
  \end{align}
  In the following let us fix $\bfxi \in C^\infty_{0,\Div}(Q)$. Let
  $$\gamma:= \bigg(\dashint_Q \abs{\nabla \bfxi}^{q'}\dxt+\dashint_Q \abs{ \bfH_\bfxi}^{q'}\dxt\bigg)^{\frac 1{q'}}$$
  and $m_0 \in \setN$, $m_0\gg1$. Due to Theorem~\ref{cor:appl} applied
  with $\sigma=q'$ we find $\lambda \in [2^{m_0}\gamma, 2^{2m_0} \gamma]$ and
  $\bfxi_\lambda \in L^\infty(4J;W^{1,\infty}_{0,\Div}(4B))$ such that
  \begin{align}
    \label{eq:xi1}
    \norm{\nabla \bfxi_\lambda}_{L^\infty(4Q)} &\leq c\, \lambda,
    \\
    \label{eq:xi2}
    \lambda^{q'} \frac{\mathcal{L}^{d+1}\left(2Q\cap\set{\bfxi_\lambda \not=
          \bfxi}\right)}{\abs{Q}} &\leq \frac{c}{m_0} \bigg(\dashint_Q
    \abs{\nabla \bfxi}^{q'}\dxt+\dashint_Q\abs{ \bfH_\bfxi}^{q'}\dxt\bigg),
    \\
    \label{eq:xi3}
    \dashint_{4Q} \abs{\bfxi_\lambda}^{q'} \dxt &\leq c
    \bigg(\dashint_Q \abs{\bfxi}^{q'} \dxt+\dashint_Q r^{q'}\abs{ \bfH_\bfxi}^{q'}\dxt\bigg),\\
       \label{eq:xi4}
    \dashint_{4Q} \abs{\nabla \bfxi_\lambda}^{q'} \dxt &\leq c
    \bigg(\dashint_Q \abs{\nabla \bfxi}^{q'} \dxt+\dashint_Q \abs{ \bfH_\bfxi}^{q'}\dxt\bigg).
    \end{align}
  Note that $\bfxi$ can be extended by 0 to $4Q$ thus the equation
  \begin{align*}
  \partial_t\bfxi=\Div\Bog_B(\partial_t\bfxi)=:\Div\bfH_{\bfxi}
  \end{align*}
  holds on $4Q$ by the properties of $\Bog_B$ (since $\bfH_\bfxi$ can be extended as well).  Theorem~\ref{cor:appl} \ref{itm:4}
      implies that $\partial_t(\bfxi-\bfxi_\lambda)\in L^{q'}(2J,W^{-1,q'}(2B))$ and
      \begin{align}\label{eq:xi5}
      \begin{aligned}
   \int_{2J} &\langle\partial_t(\bfxi-\bfxi_\lambda),\bfphi\rangle\dt\\
&   \leq c(\kappa)\int_{2Q} \chi_{\set{\bfxi\neq\bfxi_\lambda}}|\nabla\bfphi|^{q}\dxt+\kappa \bigg(\int_{Q}\abs{\nabla \bfxi}^{q'}+ \abs{ \bfH_\bfxi}^{q'}\dxt\bigg)
\end{aligned}
   \end{align}
   for all $\bfphi\in W^{1,q}_0(2Q)$.
  We calculate for $\eta\in C^\infty_0(2Q)$ with $\eta\equiv1$ on $Q$, $|\nabla^k\eta|\leq cr^{-k}$ and $|\partial_t\nabla^{k-1}\eta|\leq cr^{-(k+1)}$ ($k=1,2$)
  \begin{align*}
    \dashint_Q &\mathcal{A}(\ep(\bfv), \ep(\bfxi))\dxt -\dashint_{Q} \bfv\cdot \partial_t\bfxi\dxt \\
    &=2^{d+2}\dashint_{2Q} \mathcal{A}\big(\ep(\bfv), \ep(\eta\bfxi-\Bog_{2B\setminus B}(\nabla\eta\bfxi))\big)\dxt -\dashint_{2Q} \bfv\cdot \partial_t\big(\eta\bfxi-...\big)\dxt \\
     &=2^{d+2}\bigg(\dashint_{2Q} \mathcal{A}\big(\ep(\bfv), \ep(\eta\bfxi_\lambda-\Bog_{2B\setminus B}(\nabla\eta\bfxi_\lambda))\big)\dxt +\dashint_{2Q} \partial_t\bfv\cdot \big(\eta\bfxi_\lambda-...\big)\dxt \bigg)\\
 &+2^{d+2}\dashint_{2Q} \mathcal{A}\big(\ep(\bfv), \ep(\eta(\bfxi-\bfxi_\lambda)-\Bog_{2B\setminus B}(\nabla\eta(\bfxi-\bfxi_\lambda)))\big)\dxt\\
 &+2^{d+2}\dashint_{2Q} \partial_t\bfv\cdot \big(\eta(\bfxi-\bfxi_\lambda)-\Bog_{2B\setminus B}(\nabla\eta(\bfxi-\bfxi_\lambda))\big)\dxt\\
    &=: 2^{d+2}(I + II+III).
  \end{align*}
  Note that the time-derivative of $\bfv$ exists in the $W^{-1,\infty}_{\Div}$-sense as a consequence of (\ref{eq:almAStokespara}). Therefore all terms are well-defined by the properties of $\bfxi_\lambda$.
  We have the following inequality on account of \eqref{eq:xi3},~\eqref{eq:xi4} and the continuity properties of $\Bog$  ($\tilde{\bfxi_\lambda}:=\bfxi-\bfxi_\lambda$):
  \begin{align}\label{eq:anew}
  \begin{aligned}
 \dashint_{2Q}|\nabla\bfPsi_\lambda|^{q'}\dxt &:=\dashint_{2Q}
    \abs{\nabla(\eta\tilde{\bfxi}_\lambda)-\nabla\Bog_{2B\setminus B}( \nabla\eta\tilde{\bfxi}_\lambda)}^{q'}\dxt\\&\leq c\,\dashint_{2Q}
    \abs{\nabla \tilde{\bfxi}_\lambda}^{q'}\dxt+c\, \dashint_{2Q}
    \Big|\frac{ \tilde{\bfxi}_\lambda}{r}\Big|^{q'}\dxt\\
    &\leq  c\,\dashint_{Q}
    \abs{\nabla \bfxi}^{q'}\dxt+c\,\dashint_{Q}
    \Big|\frac{ \bfxi}{r}\Big|^{q'}\dxt+c\,\dashint_{Q}
    \abs{\bfH_\bfxi}^{q'}\dxt\\
     &\leq  c\,\dashint_{Q}
    \abs{\nabla \bfxi}^{q'}\dxt+c\,\dashint_{Q}
    \abs{\bfH_\bfxi}^{q'}\dxt.
    \end{aligned}
  \end{align}
  %We consider the terms
  %\begin{align*}
  %II_1&:=2^{d+1}\dashint_{2Q} \mathcal{A}\big(\ep(\bfv), \ep(\eta\tilde{\bfxi}_\lambda)\big)\chi_{\set{\bfxi\neq\bfxi_\lambda}}\dxt,\\
  % II_2&:=2^{d+1}\dashint_{2Q} \mathcal{A}\big(\ep(\bfv), \ep(\Bog_{2B\setminus B}(\nabla\eta(\bfxi-\bfxi_\lambda)))\big)\dxt.
  %\end{align*}
 Young's inequality for an appropriate choice of $\varepsilon>0$ and Poincar\'{e}'s inequality imply together with (\ref{eq:xi3}) and (\ref{eq:xi4})
  \begin{align*}
    II
       &\leq c(\varepsilon)\, \dashint_{2Q} \abs{\ep(\bfv)}^q \chi_{\set{\bfxi \not=
        \bfxi_\lambda}} \dxt + \varepsilon\, \dashint_{2Q}
    \abs{\nabla \bfPsi_\lambda}^{q'}\dxt\\
          &\leq c\, \dashint_{2Q} \abs{\ep(\bfv)}^q \chi_{\set{\bfxi \not=
        \bfxi_\lambda}} \dxt + \frac{1}{3}\, \dashint_{Q}
    \abs{\nabla \bfxi}^{q'}+\abs{\bfH_\bfxi}^{q'}\dxt\\
    &=: II_{1} + II_{2},
  \end{align*}
  where $c$ depends on $\abs{\mathcal{A}}$, $q$ and
  $q'$. With H{\"o}lder's inequality we gain
  \begin{align*}
    II_{1} &\leq c \bigg(\dashint_{2Q}  \abs{\nabla \bfv}^{qs}
    \dxt\bigg)^{\frac 1s} \bigg( \frac{\mathcal{L}^{d+1}\left(2Q\cap\set{\bfxi_\lambda
          \not= \bfxi}\right)}{\abs{Q}} \bigg)^{1 - \frac 1s}.
  \end{align*}
  If follows from \eqref{eq:xi2}, by the choice of~$\gamma$ and
  $\lambda \geq \gamma$ that
  \begin{align}\label{eq:kappam0}
    \frac{\mathcal{L}^{d+1}\left(2Q\cap\set{\bfxi_\lambda \not=
          \bfxi}\right)}{\abs{Q}} &\leq \frac{c \gamma^{q'}}{m_0
      \lambda^{q'}} \leq \frac{c}{m_0}.
  \end{align}
  Thus
  \begin{align*}
    II_{1} &\leq c \bigg(\dashint_{2Q} \abs{\nabla
      \bfv}^{qs}\dxt\bigg)^{\frac 1s} \bigg( \frac{c}{m_0} \bigg)^{1 -
      \frac 1s}.
  \end{align*}
  We choose $m_0$ so large such that
  \begin{align*}
    II_{1} &\leq \frac{\kappa}{3} \bigg(\dashint_{2Q} 
    \abs{\nabla \bfv}^{qs}\dxt\bigg)^{\frac 1s}.
  \end{align*}
  
  
  \iffalse
  Now we estimate $II_2$ using H\"older's inequality and the continuity of $\nabla\Bog_{2B\setminus B}$ by
  \begin{align*}
  II_2
  &\leq \bigg(\dashint_{2Q} \abs{\ep(\bfv)}^{qs}  \dxt\bigg)^{\frac{1}{qs}}\bigg(\dashint_{2Q}  |\nabla\Bog_{2B\setminus B}(\nabla\eta\tilde{\bfxi}_\lambda)|^{(qs)'}\dxt\bigg)^{\frac{1}{(qs)'}}\\
   &\leq \frac{\kappa}{3} \bigg(\dashint_{2Q} \abs{\ep(\bfv)}^{qs}  \dxt\bigg)^{\frac{1}{s}}+c_0\kappa^{-1/(q-1)}\bigg(\dashint_{2Q}  |\nabla\Bog_{2B\setminus B}(\nabla\eta\tilde{\bfxi}_\lambda)|^{(qs)'}\dxt\bigg)^{\frac{q'}{(qs)'}}\\
   &\leq \frac{\kappa}{3} \bigg(\dashint_{2Q} \abs{\ep(\bfv)}^{qs}  \dxt\bigg)^{\frac{1}{s}}+c_0\kappa^{-1/(q-1)}\bigg(\dashint_{2Q}  |\nabla\eta\tilde{\bfxi}_\lambda|^{(qs)'}\dxt\bigg)^{\frac{q'}{(qs)'}}\\
    &\leq \frac{\kappa}{3} \bigg(\dashint_{2Q} \abs{\ep(\bfv)}^{qs}  \dxt\bigg)^{\frac{1}{s}}+c_0\kappa^{-1/(q-1)}\bigg(\dashint_{2Q}  \Big|\frac{\tilde{\bfxi}_\lambda}{r}\Big|^{(qs)'}\dxt\bigg)^{\frac{q'}{(qs)'}}\\
    &=:\frac{\kappa}{3}  II_{21}+c_0\kappa^{-1/(q-1)}II_{22}.
  \end{align*}
  It holds for $m_0$ large enough on account of (\ref{eq:xi3}) and (\ref{eq:xi4}) for some $s_0>1$
    \begin{align*}
    II_1 &\leq c \bigg(\dashint_{2Q}   \Big|\frac{\tilde{\bfxi}_\lambda}{r}\Big|^{q'}
    \dxt\bigg)\bigg( \frac{\mathcal{L}^{d+1}\left(2Q\cap\set{\bfxi_\lambda
          \not= \bfxi}\right)}{\abs{Q}} \bigg)^{1 - \frac{1}{s_0}}\\
          &\leq\bigg(\dashint_{2Q}   \Big|\frac{\tilde{\bfxi}_\lambda}{r}\Big|^{q'}
    \dxt+\dashint_{2Q}  |\nabla\tilde{\bfxi}_\lambda|^{q'}
    \dxt\bigg)\bigg( \frac{c}{m_0} \bigg)^{1 -
      \frac{1}{s_0}}\\
     &\leq \tilde{\kappa}^{1/(q-1)}\bigg(\dashint_{Q}   \Big|\frac{\bfxi}{r}\Big|^{q'}
    \dxt+\dashint_{Q}  |\nabla\bfxi|^{q'}
    \dxt+\dashint_{Q}  |\bfH_\bfxi|^{q'}
    \dxt\bigg)\\
    &\leq \frac{1}{3c_0}\kappa^{1/(q-1)}\dashint_{Q}  |\nabla\bfxi|^{q'}+|\bfH_\bfxi|^{q'}\dxt.
  \end{align*}
  \fi
  
  Since $\partial_t(\bfxi-\bfxi_\lambda)\in L^{q'}(2J,W^{-1,q'}(2B))$ we can write $III$ as
  \begin{align*}
  III&=\dashint_{2Q} \bfv\cdot \partial_t\eta(\bfxi-\bfxi_\lambda)\dxt+\dashint_{2Q} \eta\bfv\cdot \partial_t(\bfxi-\bfxi_\lambda)\dxt\\
  &-\dashint_{2Q} \bfv\cdot \Bog_{2B\setminus B}(\partial_t\nabla\eta(\bfxi-\bfxi_\lambda))\dxt\\
  &-\dashint_{2Q} \Bog_{2B\setminus B}^*(\bfv)\nabla\eta\cdot\partial_t(\bfxi-\bfxi_\lambda)\dxt\\
  &=:III_1+III_2+III_3+III_4.
  \end{align*}
    The formulation in (\ref{eq:almAStokespara}) does not change if we subtract terms which are constant in space from $\bfv$. So we can assume 
\begin{align}\label{eq:meanv}    
    \dashint_{2B\setminus B}\bfv(t)\dx=0\quad\text{for a.e.}\quad t\in 2J.
\end{align}    
     Therefore $\Bog_{2B\setminus B}^*(\bfv)$ is well-defined (in the sense of $L^2_0$-duality). We consider the four terms separately and obtain
    for the first one
  \begin{align*}
  III_1&\leq \,c\,\dashint_{2I}\dashint_{2B\setminus B} \chi_{\set{\bfxi_\lambda\neq\bfxi}}\Big|\frac{\bfv}{r}\Big|\Big|\frac{\bfxi-\bfxi_\lambda}{r}\Big|\dxt\\
  &\leq c(\varepsilon)\dashint_{2I}\dashint_{2B\setminus B} \Big|\frac{\bfv}{r}\Big|^q\chi_{\set{\bfxi_\lambda\neq\bfxi}}\dxt+\varepsilon \dashint_{2Q} \Big|\frac{\bfxi-\bfxi_\lambda}{r}\Big|^{q'}\dxt\\
&=:c(\varepsilon)III_{11}+\varepsilon III_{12}.
  \end{align*}
Poincar\'{e}'s inequality and Young's inequality yield
    \begin{align*}
    III_{11} &\leq c \bigg(\dashint_{2I}\dashint_{2B\setminus B}  \Big|\frac{\bfv}{r}\Big|^{qs}
    \dxt\bigg)^{\frac 1s} \bigg( \frac{\mathcal{L}^{d+1}\left(2Q\cap\set{\bfxi_\lambda
          \not= \bfxi}\right)}{\abs{Q}} \bigg)^{1 - \frac 1s}\\
          &\leq c \bigg(\dashint_{2Q}  \abs{\nabla\bfv}^{qs}
    \dxt\bigg)^{\frac 1s} \bigg( \frac{\mathcal{L}^{d+1}\left(2Q\cap\set{\bfxi_\lambda
          \not= \bfxi}\right)}{\abs{Q}} \bigg)^{1 - \frac 1s}
  \end{align*}
  Arguing as done for the term $II_1$ yields
    \begin{align*}
    III_{11} &\leq \frac{\kappa}{12} \bigg(\dashint_{2Q} 
    \abs{\nabla \bfv}^{qs}\dxt\bigg)^{\frac 1s}.
  \end{align*}
  Moreover, we gain from (\ref{eq:xi3}) and Poincar\'{e}'s inequality
  \begin{align*}
  III_{12}&\leq c\,\dashint_{Q} \Big|\frac{\bfxi}{r}\Big|^{q'}\dxt+c\,\dashint_{Q}|\bfH_\bfxi|^{q'}\dxt\\
  &\leq c\,\dashint_{Q} |\nabla\bfxi|^{q'}\dxt+c\,\dashint_{Q}|\bfH_\bfxi|^{q'}\dxt
  \end{align*}
  and finally
    \begin{align*}
  III_1
  &\leq \frac{\kappa}{12} \bigg(\dashint_{2Q} 
    \abs{\nabla \bfv}^{qs}\dxt\bigg)^{\frac 1s}+\frac{1}{12}\bigg(\int_{Q}\abs{\nabla \bfxi}^{q'}+ \abs{ \bfH_\bfxi}^{q'}\dxt\bigg).
   \end{align*}
  As a consequence of (\ref{eq:xi5}), (\ref{eq:meanv}) and Poincar\'{e}'s inequality we obtain similarly
  \begin{align*}
  III_2&\leq c(\varepsilon)\int_{2Q} \chi_{\set{\bfxi\neq\bfxi_\lambda}}|\nabla(\eta\bfv)|^{q}\dxt+\varepsilon\bigg(\int_{Q}\abs{\nabla \bfxi}^{q'}+ \abs{ \bfH_\bfxi}^{q'}\dxt\bigg)\\
  &\leq \frac{\kappa}{12} \bigg(\dashint_{2Q} 
    \abs{\nabla \bfv}^{qs}\dxt\bigg)^{\frac 1s}+\frac{1}{12}\bigg(\int_{Q}\abs{\nabla \bfxi}^{q'}+ \abs{ \bfH_\bfxi}^{q'}\dxt\bigg)
   \end{align*}
  Taking into account continuity properties of the \Bogovskii -operator we can estimate $III_3$ via (recall (\ref{eq:meanv}))
  \begin{align*}
  III_{3}&\leq \bigg(\dashint_{2Q}\dashint_{2B\setminus B}\Big|\frac{\bfv}{r}\Big|^{qs}\dxt\bigg)^{\frac{1}{qs}}\bigg(\dashint_{2Q}r^{(qs)'}\Big|\Bog_{2B\setminus B}(\partial_t\nabla\eta(\bfxi-\bfxi_\lambda))\big)\Big|^{(qs)'}\dxt\bigg)^{\frac{1}{(qs)'}}\\
  &\leq \,c\bigg(\dashint_{2Q}|\nabla\bfv|^{qs}\dxt\bigg)^{\frac{1}{qs}}\bigg(\,\dashint_{2Q}r^{2(qs)'}\big|\partial_t\nabla\eta(\bfxi-\bfxi_\lambda)\big|^{(qs)'}\dxt\bigg)^{\frac{1}{(qs)'}}\\
   &\leq \,c\bigg(\dashint_{2Q}|\nabla\bfv|^{qs}\dxt\bigg)^{\frac{1}{qs}}\bigg(\,\dashint_{2Q}\chi_{\set{\bfxi\neq\bfxi_\lambda}}\Big|\frac{\bfxi-\bfxi_\lambda}{r}\Big|^{(qs)'}\dxt\bigg)^{\frac{1}{(qs)'}}.
  \end{align*}
We gain from Young's inequality
    \begin{align*}
  III_{3}
   &\leq \,\frac{\varepsilon^{q}}{12}\bigg(\dashint_{2Q}|\nabla\bfv|^{qs}\dxt\bigg)^{\frac 1s}+c\,\varepsilon^{-q'}\bigg(\dashint_{2Q}\chi_{\set{\bfxi\neq\bfxi_\lambda}}\Big|\frac{\bfxi-\bfxi_\lambda}{r}\Big|^{(qs)'}\dxt\bigg)^{\frac{q'}{(qs)'}}\\
   &=:\frac{\varepsilon^{q}}{12}III_{31}+c\varepsilon^{-q'}III_{32}.
  \end{align*} 
   It holds due to H\"older's inequality, (\ref{eq:kappam0}), (\ref{eq:xi2}) and (\ref{eq:xi3}) for $m_0$ large enough
   \begin{align*}
   III_{32}&\leq \bigg( \frac{\mathcal{L}^{d+1}\left(2Q\cap\set{\bfxi_\lambda
          \not= \bfxi}\right)}{\abs{Q}} \bigg)^{1 - \frac 1s}\bigg(\dashint_{2Q}\Big|\frac{\bfxi-\bfxi_\lambda}{r}\Big|^{q'}\dxt\bigg)\\
          &\leq \frac{\kappa}{12 c}\bigg(\int_{Q}\abs{\nabla \bfxi}^{q'}+ \abs{ \bfH_\bfxi}^{q'}\dxt\bigg).
   \end{align*}
   Choosing $\varepsilon:=\kappa^{1/q'}$ implies
     \begin{align*}
  III_3
  &\leq \frac{\kappa}{12 } \bigg(\dashint_{2Q} 
    \abs{\nabla \bfv}^{qs}\dxt\bigg)^{\frac 1s}+\frac{1}{12}\bigg(\int_{Q}\abs{\nabla \bfxi}^{q'}+ \abs{ \bfH_\bfxi}^{q'}\dxt\bigg).
   \end{align*}
 By (\ref{eq:xi5}) and (\ref{eq:kappam0}) we have for $m_0$ large enough
        \begin{align*}
  III_4
  &\leq  \,c\,\dashint_{2Q}\chi_{\set{\bfxi\neq\bfxi_\lambda}}
    \abs{\nabla(\nabla\eta\Bog_{2B\setminus B}^*(\bfv))}^{q}\dxt+\frac{1}{12}\bigg(\int_{Q}\abs{\nabla \bfxi}^{q'}+ \abs{ \bfH_\bfxi}^{q'}\dxt\bigg)\\
     &\leq  \,\varepsilon\,\bigg(\dashint_{2Q}\abs{\nabla(\nabla\eta\Bog_{2B\setminus B}^*(\bfv))}^{sq}\dxt\bigg)^{\frac{1}{s}}+\frac{1}{12}\bigg(\int_{Q}\abs{\nabla \bfxi}^{q'}+ \abs{ \bfH_\bfxi}^{q'}\dxt\bigg)\\
    &=:\varepsilon III_{41}+\frac{1}{2}III_{42}.
   \end{align*}
Due to the continuity of $\Bog(\Div(\cdot))$ on $L^p$ for any $1<p<\infty$ (see \cite{Ga1}[III.3, Theorem 3.3] for the \Bogovskii-operator and negative norms) we have continuity of $\nabla \Bog^\ast$ as well. 
This and and (\ref{eq:meanv}) yield
   \begin{align*}
   III_{41}&\leq \,c\,\bigg(\dashint_{2Q}\Big|\frac{\Bog_{2B\setminus B}^*(\bfv)}{r^2}\Big|^{sq}\dxt+\dashint_{2Q}\Big|\frac{\nabla\Bog_{2B\setminus B}^*(\bfv)}{r}\Big|^{sq}\dxt\bigg)^{\frac{1}{s}}\\
   &\leq \,c\,\bigg(\dashint_{2Q}\Big|\frac{\nabla\Bog_{2B\setminus B}^*(\bfv)}{r}\Big|^{sq}\dxt\bigg)^{\frac{1}{s}}
    \leq \,c\,\bigg(\dashint_{2I}\dashint_{2B\setminus B}\Big|\frac{\bfv}{r}\Big|^{sq}\dxt\bigg)^{\frac{1}{s}}\\
    &\leq \,c\,\bigg(\dashint_{2Q}|\nabla \bfv|^{sq}\dxt\bigg)^{\frac{1}{s}}
   \end{align*}
   and hence for $\varepsilon:=\kappa/12 c$
    \begin{align*}
  III_4
  &\leq \frac{\kappa}{12} \bigg(\dashint_{2Q} 
    \abs{\nabla \bfv}^{qs}\dxt\bigg)^{\frac 1s}+\frac{1}{12}\bigg(\int_{Q}\abs{\nabla \bfxi}^{q'}+ \abs{ \bfH_\bfxi}^{q'}\dxt\bigg).
   \end{align*}
 Plugging the estimates for $III_1$-$III_4$ together we see
    \begin{align*}
  III
  &\leq \frac{\kappa}{3} \bigg(\dashint_{2Q} 
    \abs{\nabla \bfv}^{qs}\dxt\bigg)^{\frac 1s}+\frac{1}{3}\bigg(\int_{Q}\abs{\nabla \bfxi}^{q'}+ \abs{ \bfH_\bfxi}^{q'}\dxt\bigg).
   \end{align*}
  Since $\bfv$ is an almost $\mathcal{A}$-Stokes solution and
  $\norm{\nabla\bfxi_\lambda}_{\infty} \leq c\, \lambda \leq c\, 2^{m_0}
  \gamma$ we have
  \begin{align*}
    |I| &\leq \delta\,\dashint_{2 \tilde{Q}} \abs{\nabla \bfv}\dxt\,
    \|\nabla \bfxi_\lambda\|_{\infty,2Q}\\
    & \leq \delta\,\bigg( \dashint_{2 \tilde{Q}} \abs{\nabla \bfv}^{qs}\dxt\bigg)^{\frac{1}{qs}}
  \, c\,2^{m_0} \gamma.
  \end{align*}
  We apply Young's inequality and Jensen's inequality to gain
  \begin{align*}
    |I| &\leq  \delta 2^{m_0}  c\bigg(\dashint_{2 \tilde{Q}} 
    \abs{\nabla \bfv}^{qs}\dxt \bigg)^{\frac 1s}+\delta 2^{m_0}  c\gamma^{q'}
    \\
    &\leq \delta 2^{m_0}  c\bigg(\dashint_{2 \tilde{Q}} 
    \abs{\nabla \bfv}^{qs}\dxt \bigg)^{\frac 1s} + \delta 2^{m_0}c
    \bigg(\dashint_{Q} \abs{\nabla \bfxi}^{q'}\dxt+\dashint_{Q} \abs{\bfH_\bfxi}^{q'}\dxt\bigg).
  \end{align*}
  Now, we choose $\delta>0$ so small such that $\delta 2^{m_0} c\leq
  \kappa/3$. Thus
  \begin{align*}
    |I| &\leq \frac{\kappa}{3} \bigg(\dashint_{2 \tilde{Q}}
    \abs{\nabla \bfv}^{qs}\dxt \bigg)^{\frac 1s} +
    \frac{1}{3}\bigg(\dashint_{Q} \abs{\nabla \bfxi}^{q'}\dxt+\dashint_{Q} \abs{\bfH_\bfxi}^{q'}\dxt\bigg).
  \end{align*}
  Combining the estimates for $I$, $II$ and $III$ we have established
  \begin{align*}
    \dashint_{2Q} \!\mathcal{A}&(\ep(\bfv) ,\ep(\bfxi))\dxt
     -\dashint_{Q} \bfv\cdot \partial_t\bfxi\dxt\\&\leq \kappa \bigg(\dashint_{2 \tilde{Q}} \abs{\nabla
      \bfv}^{qs}\dxt\bigg)^{\frac 1s} + \dashint_{Q}
    \abs{\nabla \bfxi}^{q'}\dxt+\dashint_{Q} \abs{\bfH_\bfxi}^{q'}\dxt.
  \end{align*}
Inserting this in~\eqref{eq:Liptrunvib} shows the claim
\end{proof}

\iffalse
\begin{corollary}
  \label{cor:Astokes}
 Let $\bfv\in L^{p}(2\tilde{J}; W^{1,p}_{\Div}(2\tilde{B}))$, $p\geq1$, be an
  {almost $\mathcal{A}$-Stokes solution}
 in the sense that
  \begin{align*}
    \bigg|\dashint_{2Q} \bfv\cdot \partial_t\bfxi\dxt&-\dashint_{2Q} \mathcal{A}(\ep(\bfv),\ep(\bfxi))\dxt\bigg|\leq
    \delta \bigg(\dashint_{2\tilde{Q}} |\ep(\bfv)|^{p}\dxt\bigg)^{\frac{1}{p}}\,\|\nabla\bfxi\|_\infty
  \end{align*}
  for all $\bfxi\in C^{\infty}_{0,\Div}(2Q)$ and some
  small~$\delta>0$. Let $\bfw \in
  L^p(J;W^{1,p}_{0,\Div}(B))$ be the unique solution to (\ref{eq:vH}).
Then we have
  \begin{align*}
 \dashint_{Q} \Big|\frac{\bfw}{r}\Big|^p\dxt\leq \,\kappa\,\dashint_{2\tilde{Q}}
    |\nabla\bfv|^{p}\dxt.
  \end{align*}
\end{corollary}
\begin{proof}
It is easy to see that the proof of Theorem \ref{Astokes} also applies to an almost solution in the sense of Corollary \ref{cor:Astokes}. In fact we still have
  \begin{align*}
      \dashint_{J}r_B^{-d}\|\partial_t\bfw\|^q_{W^{-1,q}_{\Div}(B)}\dt&+\dashint_{Q} \Big|\frac{\bfw}{r}\Big|^q\dxt+\dashint_{Q}
    |\nabla\bfw|^q\dxt\\&\leq \kappa(q)\bigg(\dashint_{2\tilde{Q}}
    |\nabla\bfv|^{p}\dxt\bigg)^{\frac{q}{p}}
  \end{align*}
for all $q<p$. Now, we use the interpolation
\begin{align*}
W^{1,q}(J;W^{-1,q}_{\Div})\cap L^q(J, W^{1,q}_{\Div})\hookrightarrow L^p(J,L^p_{\Div}(B))
\end{align*}
for $q$ close to $p$ and gain
  \begin{align*}
 \dashint_{Q} \Big|\frac{\bfw}{r}\Big|^p\dxt\leq \,\kappa\,\dashint_{2\tilde{Q}}
    |\nabla\bfv|^{p}\dxt.
  \end{align*}
\end{proof}


\section{Partial regularity}
In this section we study slow flows of non-stationary power-law fluids which can be described by the following set of equations in $Q:=(0,T)\times\Omega$:
\begin{align}\label{1.1'}
\left\{\begin{array}{rc}
-\partial_t \bfv+\Div \bfS=\nabla \pi-\bff& \mbox{in $Q$,}\\
\Div \bfv=0\qquad\qquad\,\,\,\,& \mbox{in $Q$,}\\
\bfv=0\qquad\,\,\,\,\quad\quad& \mbox{ \,on $\partial \Omega$,}\\
\bfv(0,\cdot)=\bfv_0\,\qquad\qquad&\mbox{ \,in $\Omega$,}\end{array}\right.
\end{align}
Here $\bfv:\Omega\rightarrow\R^d$ is the velocity, $\bfS:\Omega\rightarrow\s^d$ the stress deviator and $\pi:\Omega\rightarrow\R$ the pressure.
The quantities $\bff:Q:\rightarrow\R^d$ and $\bfv_0:\Omega\rightarrow\R^d$ are given. In contrast to the full Navier-Stokes equations (see \cite{AM} and \cite{BAH}) we neglect the convective
$(\nabla\bfv)\bfv$ which is a reasonable approprixmation if the flow is slow. In order to describe a specific fluid one needs a constitutive law which relates the stress deviator $\bfS$ and the symmetric gradient $\ep(\bfv)$ of the velocity field $\bfv$. So we assume that 
\begin{align}\label{eq:S1}
\bfS=\bfS(\ep(\bfv))
\end{align}
 where is a $\bfS\in C^1(\R^{d\times d})$. In addition we suppose
\begin{align}\label{eq:S2}
\lambda\,|\bftau|^2\leq D\bfS(\ep)(\bftau,\bftau)\leq \,\Lambda\,|\bftau|^2
\end{align}
for all $\ep,\bftau\in\s^d$ where $\lambda,\Lambda>0$. Under this assumptions it is standard to show the existence of a weak solution $\bfv\in L^\infty(0,T;L^2(\Omega))\cap L^2(0,T;W^{1,2}_{0,\Div}(\Omega))$ to (\ref{1.1'})
in the sense that
\begin{align}\label{eq:weak}
\int_Q \bfv\cdot\partial_t\bfphi\dxt-\int_Q \bfS(\ep(\bfv)):\ep(\bfphi)\dxt=\int_\Omega \bfv_0\cdot\bfphi(0,\cdot)\dx
\end{align}
for all $\bfphi\in C^\infty_{0,\Div}([0,T)\times\Omega)$. Note that we are assuming $\bff=0$ from now on since the force does not effect our regularity analysis (provided it is smooth enough).
\iffalse By DeRahms Theorem there is a a unique pressure $\pi\in L^2(0,T;L^2_0(\Omega))$ such that
\begin{align}
\begin{split}
\int_\Omega \partial_t \bfv\cdot \bfphi\,dx+\int_\Omega \bfS&(\ep(\bfv)):\ep(\phi)\,dx\\&=\int_\Omega \pi\,\Div\bfphi\,dx+\int_\Omega \bff\cdot \bfphi\,dx
\end{split}\label{23}
\end{align}
for all $\bfphi\in W_0^{1,2}(\Omega)$ and a.e. $t$. 
If we define $\bfsigma:=\bfS(\ep(\bfv))-\pi I$ then we have
\begin{align*}
\int_\Omega \partial_t \bfv\cdot \bfphi\,dx+\int_\Omega \bfsigma:\ep(\phi)\,dx=0
\end{align*}
resp. $\Div\bfsigma=\partial_t \bfv-\bff$ a.e. on $Q$.\\\fi
As a consequence of standard testing procedures with $\Delta\bfv$ and $\partial_t\bfv$ (see for instance \cite{FuS}).
\begin{lems}
\label{lems20}
Let $\bfv\in L^\infty(0,T;L^2(\Omega))\cap L^2(0,T;W^{1,2}_{0,\Div}(\Omega))$ be a solution to (\ref{eq:weak}) under the assumptions (\ref{eq:S1}) and (\ref{eq:S2}). Then we have
\begin{itemize}
\item[a)] $\bfv\in L^2(0,T;W_{loc}^{2,2}(\Omega))$;\marginpar{variational structure?}
\item[b)] $\partial_t\bfv\in L^2(0,T;L^2_{\Div}(\Omega))$;
\item[c)] $\bfS(\ep(\bfv))\in L^2(0,T;W_{loc}^{2,2}(\Omega))$.
\end{itemize}
\end{lems}
The question about regular points, i.e. points in which $\nabla\bfv$ is continuous is not well understood. Problems arise due to the interaction of the time-derivative and the pressure. The following Theorem presents are partial regularity resulkt for solutions to (\ref{1.1}) based on
the $\mathcal A$-Stokes approxmiation introduced in the last section. In addition to (\ref{eq:S2}) we need some H\"older continuity of the first derivatives of $\bfS$. To be precise we suppose
\begin{align}\label{eq:S3}
|D\bfS(\ep_1)-D\bfS(\ep_2)|\leq \,c\,|\ep_1-\ep_2|^\beta,\quad \ep_1,\ep_2\in\R^{d\times d},
\end{align}
for some $\beta\in(0,1]$.
\begin{them}
\label{thm:main}
Let $\bfv\in L^\infty(0,T;L^2(\Omega))\cap L^2(0,T;W^{1,2}_{0,\Div}(\Omega))$ be a solution to (\ref{eq:weak}) under the assumptions (\ref{eq:S1}), (\ref{eq:S2}) and (\ref{eq:S3}) . Then there is an open subset $Q_0\subset Q$ such that $\nabla\bfv\in C^\alpha(Q_0)$ and $\mathcal L^{d+1}(Q\setminus Q_0)=0$.
\end{them}
\begin{remark}
\begin{enumerate}
\item Seregin studies in \cite{Se} the full Navier-Stokes system (with the additional term $(\nabla\bfv)\bfv$ in (\ref{1.1})) under the more general power-law assumption
\begin{align*}
\lambda\,(1+|\ep|)^{p-2}|\bftau|^2\leq D\bfS(\ep)(\bftau,\bftau)\leq \,\Lambda\,(1+|\ep|)^{p-2}|\bftau|^2
\end{align*}
for $1<p<\infty$. He proves partial regularity provided
\begin{align*}
\frac{12}{5}\leq p<\frac{10}{3}.
\end{align*} 
The main reason for the lower bound $p\geq\frac{12}{5}$ is the appearance of the convective term
Seregin studies to complete system including the connective term $(\nabla\bfv)\bfv$.  However the upper bound is due to the pressure. Our technique completely avoids teh appearance of teh pressure
and we believe that it will be useful for the partial regularity of power-law fluids in the future. The main tool is a new Caccioppoli-type inequality which  is given in Theorem \ref{thm:cacc}.
%\item We believe that with some modifications our method also applies in case $1<p\leq2$. However this results can also be shown by the technique in \cite{Se}.
\item There is much more literature about the partial regularity for Non-Newtonian fluids in the stationary case. We refer to \cite{Fu4} for power-law fluids, to \cite{FuS2} for Bingham fluids and to \cite{FuS} for Prandtl-Eyring fluids. General growth assumptions on the stress daviator are supposed in \cite{BF1}, \cite{BrF} and \cite{Br}.
\end{enumerate}
\end{remark}


\begin{lemma}
\label{lems:cacc}
Assume that $\bfell=\bfP x+\bfc$ is affine linear in space and constant in time. We have for all $Q_r$ such that $2Q_r\Subset Q$ that 
\begin{align*}
\sup_{t\in I_r}\dashint_{B_r}&|\curl(\bfv-\bfell)|^2\dx+\dashint_{Q_r}|\nabla^2\bfv|^2\dxt\leq\,c\,\dashint_{Q_{2r}}\Big|\frac{\nabla\bfv-\nabla\bfell}{r}\Big|^2\dxt,
\end{align*}
for a constant $c>0$ independent of $\bfell$.
\end{lemma}
\begin{proof} We choose a cut-off function $\eta\in C^\infty_0(Q_{2r})$ with $\eta\equiv1$ on $Q_r$ and $|\nabla\eta|+|\partial_t\eta|^{\frac{1}{2}}\leq c/r$ and test (\ref{eq:weak}) with (cf. \cite{DK})
$$\bfphi=\curl\big(\eta^2\curl \big(\bfv-\bfell\big)\big),$$
where $\bfell=\bfP x+\bfc$ is affine linear in space and constant in time. We gain
\begin{align*}
\frac{d}{dt}\int_\Omega \eta^2|\curl(\bfv-\bfell)|^2\,dx
&+\int_\Omega \eta^2\curl\big\{\bfS(\ep(\bfv))\big\}:\nabla\curl\bfv\,dx\\
=&-\int_\Omega \curl\big\{\bfS(\ep(\bfv))\big\}:\nabla \eta^2\odot(\partial_\gamma\bfv-\bfP_\gamma)\,dx\\
&-\int_\Omega \partial_t\eta^2|\curl(\bfv-\bfell)|^2\,dx\\
\Longleftrightarrow: (I)&+(II)=(III)+(IV).
\end{align*}
The growth of $\bfS$ in \eqref{eq:S2} yields
\begin{align*}
|(II)^m|&\geq \,c\,\int_{B_{2r}}\eta^2|\nabla\ep(\bfv)|^2\dx\geq \,c\,\int_{B_{2r}}\eta^2|\nabla^2\bfv|^2\dx
\end{align*}
as well as
\begin{align*}
|(III)|&\leq \,c\,\int_{B_{2r}}\eta|\nabla\ep(\bfv)||\nabla\eta||\nabla(\bfv_m-\bfell)|\dx\\
&\leq \,\delta\,\int_{B_{2r}}\eta^2|\nabla\ep(\bfv)|^2\dx+c(\delta)\int_{B_{2r}}\Big|\frac{\nabla(\bfv_m-\bfell)}{r}\Big|^2\dx.
\end{align*}
Here we used Young's inequality for $\delta>0$ arbitrary. Finally we have
\begin{align*}
|(IV)|&\leq \,c\,\int_{B_{2r}} \Big|\frac{\nabla(\bfv-\bfell)}{r}\Big|^2\,dx.
\end{align*}
Now, integrating with respect to time and choosing $\delta$ small enough yields the claim of a).
\end{proof}

\begin{them}
\label{thm:cacc}
Assume that $\bfell=\bfP x+\bfc$ is affine linear in space and constant in time.
Then we have for all $Q_r$ such that $2Q_r\Subset Q$ that
\begin{align*}
\dashint_{Q_{r}}|\nabla\bfv-\nabla\bfell|^2\dxt&\leq \,c\,\dashint_{Q_{2r}}\Big|\frac{\bfv-\bfell}{r}\Big|^2\dxt.
\end{align*}
\end{them}
\begin{proof}
The proof of b) mainly relies on the following nonlinear interpolation argument: Let $\bfu\in W^{2,2}(B)$, where $B\subset\R^d$ is a ball. Then we have
\begin{align}\label{eq:int}
\int_B |\nabla\bfu|^2\dx&\leq\,\Theta \,r_B^2\,\int_B |\nabla^2\bfu|^2\dx+
c(\Theta)\,\int_B \Big|\frac{\bfu}{r_B}\Big|^2\dx
\end{align}
for all $\Theta>0$. Applying this for $\bfu=\bfv-\bfell$ and using Lemma \ref{lems:cacc} yields
\begin{align*}
\int_{Q_r} |\nabla\bfv-\nabla\bfell|^2\dxt&\leq\,\frac{1}{2} \,\int_{Q_{2r}} |\nabla\bfv-\nabla\bfell|^2\dxt+
c\,\int_{Q_r} \Big|\frac{\bfv-\bfell}{r}\Big|^2\dxt.
\end{align*}
Now, using ??? allows us to get rid of the first term on the right hand side and the claim follows.
\end{proof}

Now we introduce the excess functional
\begin{align*}
\Phi(t_0,x_0,r,\bfA):=\dashint_{Q_r(t_0,x_0)}|\nabla\bfv-\bfA|^2\dxt,\quad \bfA\in\R^{d\times d}
\end{align*}
and show that $\bfu:=\bfv-\bfA x$ is an almost solution provided $\Phi$ is small.
\begin{lems}
\label{lems3.2} Under the assumptions of Theorem \ref{thm:main} we have for $\bfA\in \R^{d\times d}$ arbitrary and all $\bfphi\in C^\infty_{0,\Div}(Q_r)$
\begin{align*}
\bigg|\dashint_{Q_r}\bfu\cdot\partial_t\bfphi\dxt&-\dashint_{Q_r}D\bfS(\bfA)(\ep(\bfu),\ep(\bfphi))\,dx\bigg|\\&\leq \,
c\,\Phi(t_0,x_0,r)^{\frac{\beta}{2}}\bigg(\dashint_{Q_r}|\nabla\bfu|^2\dxt\bigg)^{\frac{1}{2}}\|\nabla\phi\|_\infty.
\end{align*}
Here we set $\bfu=\bfv-\bfA x$.
\end{lems}
\begin{proof} The equation clearly yields
\begin{align*}
\bigg|\dashint_{Q_r}&\bfu\cdot\partial_t\bfphi\dxt-\dashint_{Q_r}D\bfS(\bfA)(\ep(\bfu),\ep(\bfphi))\dxt\bigg|\\
&=\bigg|\dashint_{Q_r}\Big[D\bfS(\bfA)(\ep(\bfu),\ep(\bfphi))+\bfS(\bfA):\ep(\bfphi)-\bfS(\ep(\bfv)):\ep(\bfphi)\Big]\dxt\bigg|
\end{align*}
Now we estimate using (\ref{eq:S3})
\begin{align*}
&|D\bfS(\bfA)(\ep(\bfu),\ep(\bfphi))+\bfS(\bfA):\ep(\bfphi)-\bfS(\ep(\bfv)):\ep(\bfphi)|\nonumber\\
&\leq \int_0^1|D\bfS(\bfA)-D\bfS(\bfA+t\nabla \bfu)|\,dt|\nabla \bfu|\|\nabla\bfphi\|_\infty\nonumber\\
&\leq \,c\,|\nabla \bfu|^{\beta}|\nabla \bfu|\|\nabla\bfphi\|_\infty.
\end{align*}
This means we need to estimate the term $\mathscr T:=\dashint_{Q_r}|\nabla \bfu|^{\frac{\beta}{2}}|\nabla \bfu|\dxt$ and gain by H\"older's inequality and Jensen's inequality
\begin{align*}
\mathscr T
&\leq\,c\,\bigg(\dashint_{Q_r}|\nabla \bfu|^{2\beta}\dxt\bigg)^{\frac{1}{2}}\bigg(\dashint_{Q_r}|\nabla \bfu|^p\dxt\bigg)^{\frac{1}{p}}\\
&\leq\,c\,\bigg(\dashint_{Q_r}|\nabla \bfu|^{2}\dxt\bigg)^{\frac{\beta}{2}}\bigg(\dashint_{Q_r}|\nabla \bfu|^p\dxt\bigg)^{\frac{1}{p}}\\
&=\,c\, \Phi(t_0,x_0,r)^{\frac{\beta}{2}}\bigg(\dashint_{Q_r}|\nabla\bfu|^2\dxt\bigg)^{\frac{1}{2}}.
\end{align*}
This proves the claim.
\end{proof}

The claim of Theorem \ref{thm:main} is a consequence of an iteration of the following Proposition and the choice
\begin{align*}
Q_0:=\set{(t_0,x_0)\in Q:\,\,\liminf_{r\rightarrow0}\Phi (t_0,x_0,r)=0,\,\,\limsup_{r\rightarrow0}|(\nabla\bfv)_{Q_{r}(t_0,x_0)}|<\infty}.
\end{align*}
\begin{props}
\label{LEMS} Let the assumptions of Theorem \ref{thm:main} hold. Let us
consider $L>0$ and $\beta\in(0,1/2)$. Then there is
$\delta=\delta(L,\beta)$ and $\theta\in (0,1)$ such that
\begin{equation}
\Phi (t_0,x_0,\rho):=\Phi (t_0,x_0,\rho, (\nabla\bfv)_{x_0,\varrho})\leq\delta,\quad |(\nabla\bfv)_{x_0,\rho}|\leq
L\label{LEM}
\end{equation}
imply
$$\Phi (t_0,x_0,\theta\rho)\leq \theta^{2\beta}\Phi (t_0,x_0,\rho).$$
\end{props}
\begin{proof} We define $\bfA:=(\nabla\bfv)_{x_0,\rho}$ so that
$|\bfA|\leq L$, $\bfu=\bfv-\bfA x$ and $\delta:=c\,\Phi(x_0,\rho)\big)^{\beta/2}\ll1$. Since
the case $\Phi(x_0,\rho)=0$ is trivial, we can assume
$\Phi(x_0,\rho)\neq0$.
We will use the $\mathcal{A}$-Stokes approximation from Corollary \ref{Astokes}, where
$\mathcal{A}:=D \bfS(\bfA)$. On account of Lemma \ref{lems3.2} the function $\bfu$ is an almost solution to the $\mathcal A$-Stokes problem in the sense of
Corollary \ref{Astokes}. We obtain a solution $\bfh\in C^\infty_{\Div}(Q_{\rho/2})$ to the $\mathcal{A}$-Stokes problem with boundary data $\bfu$. Defining $s:=\Phi(x_0,\rho)^{\frac{1}{2}}\leq1$ we gain\marginpar{check!}
\begin{align}\label{LEM3}
\sup_{Q_{\rho/2}}|\nabla \bfh|+&\rho \sup_{Q_{\rho/2}}|\nabla^2 \bfh|+\rho \sup_{Q_{\rho/2}}|\partial_t\bfh|\leq C^\ast s \\
\label{LEM3'}& \fint_{Q_{\rho}}\Big|\frac{\bfu-\bfh}{\rho}\Big|^2\dx\leq s^2\kappa.
\end{align}
Inequality (\ref{LEM3}) needs some explanation ???. 
For $\theta\in(0,\tfrac{1}{4}]$ from (\ref{LEM3}) we deduce by Taylor expansion
\begin{align}\label{neu*}
\begin{aligned}
\sup_{B_{2\theta\rho}(x_0)}&|\bfh(t_0,x)-\nabla\bfh(t_0,x)(x-x_0)-\bfh(t_0,x_0)|\leq c\theta^2 \rho s,\\
&\sup_{Q_{2\theta\rho}(x_0)}|\bfh(t,x)-\bfh(t_0,x)|\leq c\theta^2 \rho s.
\end{aligned}
\end{align}
The Caccioppoli
inequality from Theorem \ref{thm:cacc} yields
\begin{align}
&\Phi(x_0,\theta\rho,\bfA)\leq \,\Phi(x_0,\theta\rho,\bfA+\nabla \bfh(x_0))\nonumber\\&\leq\,c\,\dashint_{Q_{2\theta\rho}}\Big|\frac{\bfu-\nabla\bfh(t_0,x_0)(x-x_0)-\bfh(t_0,x_0)}{\theta\rho}\Big|^2\dxt\label{LEM6}\\&\leq \,c\,\dashint_{Q_{2\theta\rho}}\Big|\frac{\bfu-\bfh}{\theta\rho}\Big|^2\dxt+\,c\,\dashint_{Q_{2\theta\rho}}\Big|\frac{\bfh-\nabla\bfh(t_0,x_0)(x-x_0)-\bfh(t_0,x_0)}{\theta\rho}\Big|^2\dxt\nonumber
\end{align}
Combining (\ref{LEM3}) and (\ref{neu*}) shows
\begin{align}
&\Phi(x_0,\theta\rho,\bfA)\leq K(\theta)\tfrac{s^2\kappa}{\theta^p}+cs^2\theta^2.\label{LEM7}
\end{align}
Setting we define $\kappa:=\theta^{2p}/K(\theta)$ we have finally shown
\begin{align*}
\Phi(x_0,\theta\rho)\leq \theta^{2\beta}\Phi(x_0,\rho)
\end{align*}
if $\theta$ is small enough.
\end{proof}
\fi


\subsection*{Acknowledgement}
\begin{itemize}
\item The work of the author was partially supported by Leopoldina (German National Academy of Science).
\item The author wishes to thank R. Farwig for very helpful advices regarding the $L^q$-theory for the Stokes system.
\end{itemize}
%\bibliographystyle{amsalpha}
%\bibliography{lars}




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[BrDF]{BrDF D. Breit, L. Diening, M. Fuchs (2012): Solenoidal Lipschitz truncation and applications in fluid mechanics. J. Diff. Eq. 253, 1910--1942. %
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[BrVe]{BrVe D. Breit, A. Verde (2011): Quasiconvex variational integrals in Orlicz-Sobolev spaces. To appear in Ann. Mat. Pura Appl., DOI: 10.1007/s10231-011-0222-1.
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[BrenS]{BreSco94 S.~C. Brenner and L.~R. Scott (1994): The mathematical theory of finite element methods. Texts in Applied Mathematics, vol.~15, Springer-Verlag, New York. %
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[BrD]{BrD D. Breit, L. Diening (2011): Sharp conditions for Korn inequalities in Orlicz spaces. Preprint 290, Saarland University. %
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[CrKr]{CrKr D. Cruz-Uribe, M. Krbec (2002): Localization and extrapolation in Lorentz-Orlicz spaces. Function Spaces, Interpolation Theory and Related Topics, M. Cwikel et al. Eds., de Gruyter, Berlin, 273-284. %
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[CE]{CE M. Chipot, L. C. Evans (1986): Linearization at infinity and Lipschitz estimates for certain problems in the calculus of variations. Proc. Royal Soc. Edinburgh, Sect. A, 102, 291--303.
1402.3064.extracted_bib
[ByWa]{ByWa S.-S. Byun, L. Wang (2005): $L^p$ estimates for parabolic equations in Reifenberg domains. J. Funct. Anal. 223, 44--85.
1402.3064.extracted_bib
[DK]{DK L. Diening, P. Kaplicky (2012): $L^q$ theory for a generalized Stokes System. Manuscripta Mathematica 141, Issue 1-2, 333--361.
1402.3064.extracted_bib
[DieR2]{DieR07 L.~Diening and M.~R{\r u{\v zi{\v cka (2007): Interpolation operators in {Orlicz~{Sobolev spaces. Num. Math. 107 no.~1, 107--129.
1402.3064.extracted_bib
[De1]{DeG61 E. De Giorgi (1961): Frontiere orientate di misura minima, Seminario di Matematica della Scuola Normale Superiore di Pisa, 1960-61, Editrice Tecnico Scientica, Pisa. %
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[De2]{De2 E. DeGiorgi (1957): Sulla differenziabilit\'{a e l'analiticit\'{a delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino 3, 25--43.
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[DM1]{DM1 F. Duzaar and G. Mingione, (2004): The p-harmonic approximation and the regularity of p-harmonic maps. Calc. Var. Partial Differential Equations, 20 (2004), pp. 235--256.
1402.3064.extracted_bib
[DM2]{DuzMin09h F. Duzaar and G. Mingione (2009): Harmonic type approximation lemmas, J. Math. Anal. Appl. 352, no. 1, 301--335.
1402.3064.extracted_bib
[DuzM2]{DM05 F. Duzaar, G. Mingione (2005): Second order parabolic systems, optimal regularity, and singular sets of solutions. Ann. Inst. Henri Poincar\'{e, Anal. Non Lin\'{eaire 22, 705--751. %
1402.3064.extracted_bib
[DMM]{DalMasoMurat G.~Dal~Maso and F.~Murat (1998): %\newblock Almost everywhere convergence of gradients of solutions to nonlinear elliptic systems. %\newblock {\em Nonlinear Anal., 31(3-4), 405--412.
1402.3064.extracted_bib
[DieLSV]{DLSV L. Diening, B. Stroffolini, A. Verde (2012): Partial regularity for minimizers of quasiconvex functionals with general growth. SIAM J. Math. Anal. 44, 3594--3616.
1402.3064.extracted_bib
[DMS]{DMS L. Diening, J. Mal{\'ek, M. Steinhauer (2008): On Lipschitz Truncations of Sobolev Functions (with Variable Exponent) and Their Selected Applications. ESAIM: Control, Optimisation and Calculus of Variations 14, 211--232. %
1402.3064.extracted_bib
[DiKrSu]{DiKrSu L. Diening, Ch. Kreuzer, E. S\"uli: Finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology. 2012. Submitted.
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[DuMiSt]{DuMiSt F. Duzaar, R. Mingione, K. Steffen (2011): Parabolic Systems with Polynomial Growth and Regularity. Memoires of the American Mathematical Society 214, no. 1005.
1402.3064.extracted_bib
[DSV]{DSV L. Diening, B. Stroffolini, A. Verde (2012): The $\varphi$--harmonic approximation and the regularity of $\varphi$--harmonic maps, J. Diff. Eq. 253, 1943--1958.
1402.3064.extracted_bib
[DRW]{DRW L. Diening, M. R\r{u\v{zi\v{cka, J. Wolf (2010): Existence of weak solutions for unsteady motions of generalized Newtonian fluids. Ann. Sc. Norm. Sup. Pisa Cl. Sci. (5) Vol. IX (2010), 1-46 %
1402.3064.extracted_bib
[E]{E H. J. Eyring (1936): Viscosity, plasticity, and diffusion as example of absolute reaction rates. J. Chemical Physics 4, 283-291. %
1402.3064.extracted_bib
[Fu1]{Fu1 M. Fuchs (2008): Regularity results for local minimizers of energies with general densities having superquadratic growth. to appear in Algebra i Analysis/ Preprint 217, Saarland University. %
1402.3064.extracted_bib
[ELM]{ELM L. Esposito, F. Leonetti, G. Mingione (2001): Sharp regularity for functionals with $(p,q$)-growth. Journal of Differential Equations 204, 5--55.
1402.3064.extracted_bib
[FarKS]{Fa R. Farwig, H. Kozono, H. Sohr (2005): An $L^q$-approach to Stokes and Navier-Stokes equations in general domains. Acta Math. 195, 21--53.
1402.3064.extracted_bib
[Fu2]{Fu4 M. Fuchs (1996): On quasistatic Non-Newtonian fluids with power law. Math. Meth. Appl. Sciences 19, 1225-1232.
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[Ga1]{Ga1 G. Galdi (1994): An introduction to the mathematical theory of the Navier-Stokes equations Vol. I, Springer Tracts in Natural Philosophy Vol. 38. Springer, Berlin-New York.
1402.3064.extracted_bib
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