Regularity for parabolic systems of Uhlenbeck type with Orlicz growth

TL;DR

This paper establishes new local regularity results for solutions to non-linear parabolic systems with Orlicz growth, extending previous work on p-Laplace equations by developing advanced gradient estimates.

Contribution

It introduces novel local gradient estimates for parabolic systems with Orlicz growth, bridging a gap in regularity theory for Uhlenbeck-type systems.

Findings

Proved local Hölder continuity of solutions with bounded gradients.

Generalized and improved estimates for the gradient of solutions.

Extended regularity results to systems with Orlicz growth structures.

Abstract

We study the local regularity of $p$-caloric functions or more generally of $\phi$-caloric functions. In particular, we study local solutions of non-linear parabolic systems with homogeneous right hand side, where the leading terms has Uhlenbeck structure of Orlicz type. This paper closes the gap of [22] where Liebermann proved that if the gradient of a solution is bounded, it is H\"older continuous. The crucial step is a novel local estimates for the gradient of the solutions, which generalize and improve the pioneering estimates of DiBenedetto and Friedman [12,10] for the $p$-Laplace heat equation.