Regularity theory for parabolic systems with Uhlenbeck structure

TL;DR

This paper develops a local regularity theory for parabolic systems with Uhlenbeck structure and $\

Contribution

It introduces a unified approach to establish regularity results for both degenerate and singular parabolic systems with $\\varphi$-growth.

Findings

Proved local boundedness of solutions and their gradients.

Established local Hölder continuity of gradients.

Applicable to a broad class of systems with different degeneracy levels.

Abstract

We establish local regularity theory for parabolic systems of Uhlenbeck type with $\varphi$-growth. In particular, we prove local boundedness of weak solutions and their gradient, and then local H\"older continuity of the gradients, providing suitable assumptions on the growth function $\varphi$. Our approach, being independent of the degeneracy of the system, allows for a unified treatment of both the degenerate and the singular case.