Partial regularity for type two doubly nonlinear parabolic systems

We study weak solutions $\mathbf{v}:U\times (0,T)\to {\mathbb{R}}^m$ of the nonlinear parabolic system $$D\psi ({\mathbf{v}}_t)=\text{div}DF(D\mathbf{v}),$$ 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 $D^2\mathbf{v}$ and ${\mathbf{v}}_t$ 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 $D^2\mathbf{v}$ and ${\mathbf{v}}_t$.