Partial gradient regularity for parabolic systems with degenerate diffusion and H\"older continuous coefficients

TL;DR

This paper proves that the gradient of weak solutions to certain degenerate or singular parabolic systems with H"older continuous coefficients exhibits partial H"older continuity, extending regularity results beyond Uhlenbeck-type structures.

Contribution

It establishes partial gradient regularity for degenerate parabolic systems with H"older continuous coefficients, even when the vector field lacks Uhlenbeck-type structure.

Findings

Gradient $Du$ is partially H"older continuous under specified conditions.

Regularity holds for systems with $p$-growth and H"older continuous coefficients.

Results extend regularity theory to more general degenerate parabolic systems.

Abstract

We consider vector valued weak solutions $u:\Omega_T\to \mathbb{R}^N$ with $N\in \mathbb{N}$ of degenerate or singular parabolic systems of type \begin{equation*} \partial_t u - \mathrm{div} \, a(z,u,Du) = 0 \qquad\text{in}\qquad \Omega_T= \Omega\times (0,T), \end{equation*} where $\Omega$ denotes an open set in $\mathbb{R}^{n}$ for $n\geq 1$ and $T>0$ a finite time. Assuming that the vector field $a$ is not of Uhlenbeck-type structure, satisfies $p$-growth assumptions and $(z,u)\mapsto a(z,u,\xi)$ is H\"older continuous for every $\xi\in \mathbb{R}^{Nn}$, we show that the gradient $Du$ is partially H\"older continuous, provided the vector field degenerates like that of the $p$-Laplacian for small gradients.