Partial regularity for type two doubly nonlinear parabolic systems

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 ${\bf v}_t$.