Schauder estimates for parabolic $p$-Laplace systems

TL;DR

This paper proves local Hölder regularity of the spatial gradient for solutions to a class of nonlinear parabolic systems, extending regularity results to systems with non-constant coefficients and applications to doubly nonlinear equations.

Contribution

It establishes new Hölder regularity results for the gradient of solutions to nonlinear parabolic systems with variable coefficients, including applications to super-critical fast diffusion equations.

Findings

Proved local Hölder continuity of the spatial gradient of solutions.

Extended regularity results to systems with Hölder continuous coefficients.

Applied findings to doubly nonlinear parabolic equations in the super-critical regime.

Abstract

We establish the local H\"older regularity of the spatial gradient of bounded weak solutions $u\colon E_T\to\R^k$ to the non-linear system of parabolic type \begin{equation*} \partial_tu-\Div\Big( a(x,t)\big(\mu^2+|Du|^2\big)^\frac{p-2}2Du\Big)=0 \qquad\mbox{in $E_T$}, \end{equation*} where $p>1$, $\mu\in[0,1]$, and the coefficient $a\in L^\infty(E_T)$ is bounded below by a positive constant and is H\"older continuous in the space variable $x$. As an application, we prove H\"older estimates for the gradient of weak solutions to a doubly non-linear parabolic equation in the super-critical fast diffusion regime.