The A-Stokes approximation for non-stationary problems

Abstract

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_t\phi\,dx\,dt-\int_Q \mathcal A(\varepsilon(v),\varepsilon(\phi))\,dx\,dt=0\quad\forall\phi\in C^{\infty}_{0,div}(Q), \end{align} where $Q:=I\times B$, $B\subset\mathbb R^d$ 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 $v\in L^p(I;W^{1,p}_{div}(B))$, $1<p<\infty$, can be approximated by a solution in the $L^s(I;W^{1,s}(B))$-sense for all $s<p$. So, we extend the stationary $\mathcal A$-Stokes approximation by Breit-Diening-Fuchs to parabolic problems.