Compactness methods for doubly nonlinear parabolic systems

Hynd, Ryan

doi:10.1090/tran/6828

Compactness methods for doubly nonlinear parabolic systems
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Trans. Amer. Math. Soc. 369 (2017), 5031-5068 Request permission

Abstract:

We study solutions of the system of PDE $D\psi (\textbf {v}_t)=\mathrm {Div} DF(D\textbf {v})$, 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.

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