On regularity of the time derivative for degenerate parabolic systems

TL;DR

This paper establishes regularity estimates for the time derivatives of a broad class of nonlinear parabolic systems, including models for non-Newtonian fluids, using innovative weak difference quotients.

Contribution

It introduces novel regularity estimates for time derivatives in nonlinear parabolic systems, applicable to complex models like p-Laplace and non-Newtonian fluids.

Findings

Bounded fractional derivatives of time derivatives in various systems

Applicable to p-Laplace and non-Newtonian fluid models

Estimates hold under very general assumptions

Abstract

We prove regularity estimates for time derivatives of a large class of nonlinear parabolic partial differential systems. This includes the instationary (symmetric) p-Laplace system and models for non Newtonien fluids of powerlaw or Carreau type. By the use of special weak different quotients, adapted to the variational structure we bound fractional derivatives of $u_t$ in time and space direction. Although the estimates presented here are valid under very general assumptions they are a novelty even for the parabolic p-Laplace equation.