Partial regularity for doubly nonlinear parabolic systems of the first type

We study solutions $\mathbf{v}$ of the parabolic system of PDE $${\mathrm{\partial }}_t(D\psi (\mathbf{v}))=\text{div}DF(D\mathbf{v})\mathrm{.}$$ Here $\psi$ and $F$ are convex functions, and this is a model equation for more general doubly nonlinear evolutions that arise in the study of phase transitions in materials. We show that if $\mathbf{v}$ is a weak solution, then $D\mathbf{v}$ is locally H"older continuous except for possibly on a lower dimensional subset of the domain of $\mathbf{v}$. Our proof is based on compactness properties of solutions, two integral identities and a fractional time derivative estimate for $D\mathbf{v}$.