Compactness methods for doubly nonlinear parabolic systems
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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.References
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Additional Information
- Ryan Hynd
- Affiliation: Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 19104
- MR Author ID: 789875
- Received by editor(s): December 6, 2014
- Received by editor(s) in revised form: August 23, 2015
- Published electronically: December 27, 2016
- Additional Notes: The author was partially supported by NSF grant DMS-1301628
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 5031-5068
- MSC (2010): Primary 35B65, 35K40, 35K55, 35P30
- DOI: https://doi.org/10.1090/tran/6828
- MathSciNet review: 3632559