Degenerate nonlinear parabolic equations with discontinuous diffusion coefficients

Kwon, Dohyun; Mészáros, Alpár Richárd

doi:10.1112/jlms.12444

Degenerate nonlinear parabolic equations with discontinuous diffusion coefficients

Kwon, Dohyun; Mészáros, Alpár Richárd

Authors

Dohyun Kwon

Dr Alpar Meszaros alpar.r.meszaros@durham.ac.uk
Associate Professor

Abstract

This paper is devoted to the study of some nonlinear parabolic equations with discontinuous diffusion intensities. Such problems appear naturally in physical and biological models. Our analysis is based on variational techniques and in particular on gradient flows in the space of probability measures equipped with the distance arising in the Monge–Kantorovich optimal transport problem. The associated internal energy functionals in general fail to be differentiable, therefore classical results do not apply directly in our setting. We study the combination of both linear and porous medium type diffusions and we show the existence and uniqueness of the solutions in the sense of distributions in suitable Sobolev spaces. Our notion of solution allows us to give a fine characterization of the emerging critical regions, observed previously in numerical experiments. A link to a three phase free boundary problem is also pointed out.

Citation

Kwon, D., & Mészáros, A. R. (2021). Degenerate nonlinear parabolic equations with discontinuous diffusion coefficients. Journal of the London Mathematical Society, 104(2), 688-746. https://doi.org/10.1112/jlms.12444

Journal Article Type	Article
Acceptance Date	Feb 6, 2021
Online Publication Date	Feb 22, 2021
Publication Date	Sep 1, 2021
Deposit Date	Feb 22, 2021
Publicly Available Date	Oct 7, 2021
Journal	Journal of the London Mathematical Society
Print ISSN	0024-6107
Electronic ISSN	1469-7750
Publisher	Wiley
Peer Reviewed	Peer Reviewed
Volume	104
Issue	2
Pages	688-746
DOI	https://doi.org/10.1112/jlms.12444
Related Public URLs	https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.12444

Files

Published Journal Article (708 Kb)
PDF

Publisher Licence URL
http://creativecommons.org/licenses/by-nc-nd/4.0/

Copyright Statement
© 2021 The Authors. Journal of the London Mathematical Society is copyright © London Mathematical Society.

This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.