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arXiv · 1702.00537

Partial regularity for doubly nonlinear parabolic systems of the first type

Ryan Hynd
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Abstract

We study solutions ${\bf v}$ of the parabolic system of PDE $$ \partial_t\left(Dψ({\bf v})\right)=\text{div}DF(D{\bf v}). $$ Here $ψ$ 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 ${\bf v}$ is a weak solution, then $D{\bf v}$ is locally Hölder continuous except for possibly on a lower dimensional subset of the domain of ${\bf v}$. Our proof is based on compactness properties of solutions, two integral identities and a fractional time derivative estimate for $D{\bf v}$.

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Ryan Hynd. 2018-04-25. Partial regularity for doubly nonlinear parabolic systems of the first type. https://arxiv.org/abs/1702.00537

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