Degenerate nonlinear parabolic equations with discontinuous diffusion coefficients

Kwon, Dohyun; Mészáros, Alpár Richárd

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
• Public URL: https://durham-repository.worktribe.com/output/1252239
• Related Public URLs: https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.12444