Everywhere regularity of certain types of parabolic systems

TL;DR

This paper proves that bounded weak solutions of certain nonlinear parabolic systems with convex potential are everywhere Holder continuous and smooth, extending regularity results and simplifying existing proofs for coupled systems.

Contribution

It establishes the everywhere regularity of solutions for a class of parabolic systems with convex potential and simplifies the proof of known regularity results for strongly coupled systems.

Findings

Bounded weak solutions are everywhere Holder continuous.

Solutions are everywhere smooth under specified conditions.

Method simplifies existing proofs of regularity for coupled systems.

Abstract

In this paper I discuss nonlinear parabolic systems that are generalizations of scalar diffusion equations. I show that when potential is a convex function that depends only on the norm of the solution, then bounded weak solutions of these parabolic systems are everywhere Holder continuous and thus everywhere smooth. I also show that the method used to prove this result can be easily adopted to simplify the proof of the result due to Wiegner on everywhere regularity of bounded weak solutions of strongly coupled parabolic systems.