Natural second-order regularity for parabolic systems with operators having $(p,\delta)$-structure and depending only on the symmetric gradient

TL;DR

This paper introduces a new approximation method to establish global second-order regularity results for parabolic systems with $(p, ext{delta})$-structure operators depending on the symmetric gradient, applicable for all $p>1$ and $ ext{delta}>0$.

Contribution

The paper develops a novel approximation technique enabling energy methods for regularity results in parabolic systems with general potential operators, extending known results and providing new insights.

Findings

Proves natural second order regularity up to the boundary for $p>2$.

Method applies to elliptic and parabolic cases, including $1<p extless=2$.

Provides a unified approach to regularity for operators with $(p, ext{delta})$-structure.

Abstract

In this paper we consider parabolic problems with stress tensor depending only on the symmetric gradient. By developing a new approximation method (which allows to use energy-type methods typical for linear problems) we provide an approach to obtain global regularity results valid for general potential operators with $(p,\delta)$-structure, for all $p>1$ and for all $\delta>0$. In this way we prove ``natural'' second order spatial regularity -- up to the boundary -- in the case of homogeneous Dirichlet boundary conditions. The regularity results, are presented with full details for the parabolic setting in the case $p>2$. However, the same method also yields regularity in the elliptic case and for $1<p\leq 2$, thus proving in a different way results already known.