Compactness methods for doubly nonlinear parabolic systems

TL;DR

This paper investigates the mathematical properties of solutions to a class of doubly nonlinear parabolic PDE systems, establishing compactness, partial regularity, and large-time behavior, with implications for physical models of phase transitions.

Contribution

It introduces new compactness and regularity results for solutions of doubly nonlinear parabolic systems with convex energy functions.

Findings

Established compactness properties of solutions.

Verified partial regularity for quadratic systems.

Characterized large time limits of weak solutions.

Abstract

We study solutions of the system of PDE $D\psi({\bf v}_t)=\text{div}DF(D{\bf v})$, where $\psi$ and $F$ are convex functions. This type of system arises in various physical models for phase transitions. We establish compactness properties of solutions that allow us to verify partial regularity when $F$ is quadratic and characterize the large time limits of weak solutions. Special consideration is also given to systems that are homogeneous and their connections with nonlinear eigenvalue problems. While the uniqueness of weak solutions of such systems of PDE remains an open problem, we show scalar equations always have a preferred solution that is also unique as a viscosity solution.