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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Compactness methods for doubly nonlinear parabolic systems


Author: Ryan Hynd
Journal: Trans. Amer. Math. Soc. 369 (2017), 5031-5068
MSC (2010): Primary 35B65, 35K40, 35K55, 35P30
DOI: https://doi.org/10.1090/tran/6828
Published electronically: December 27, 2016
MathSciNet review: 3632559
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Abstract: We study solutions of the system of PDE $ D\psi ({\bf v}_t)=\mathrm {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.


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Additional Information

Ryan Hynd
Affiliation: Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 19104

DOI: https://doi.org/10.1090/tran/6828
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
Article copyright: © Copyright 2016 American Mathematical Society

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