# Partial regularity for type two doubly nonlinear parabolic systems

## TL;DR

This paper proves partial regularity results for solutions to a class of doubly nonlinear parabolic systems, showing that second derivatives and time derivatives are locally Hölder continuous outside a lower-dimensional set.

## Contribution

It establishes partial regularity for weak solutions of doubly nonlinear parabolic systems with Hölder continuous second derivatives of the nonlinear function, extending understanding of their structural properties.

## Key findings

- Second derivatives and time derivatives are locally Hölder continuous outside a lower-dimensional set.
- The proof uses integral identities, energy decay, and fractional derivative estimates.
- Results apply to models in material science involving doubly nonlinear evolutions.

## Abstract

We study weak solutions ${\bf v}:U\times (0,T)\rightarrow \mathbb{R}^m$ of the nonlinear parabolic system $$ D\psi({\bf v}_t)=\text{div}DF(D{\bf v}), $$ where $\psi$ and $F$ are convex functions. This is a prototype for more general doubly nonlinear evolutions which arise in the study of structural properties of materials. Under the assumption that the second derivatives of $F$ are H\"older continuous, we show that $D^2{\bf v}$ and ${\bf v}_t$ are locally H\"older continuous except for possibly on a lower dimensional subset of $U\times (0,T)$. Our approach relies on two integral identities, decay of the local space-time energy of solutions, and fractional time derivative estimates for $D^2{\bf v}$ and ${\bf v}_t$.