Degenerate nonlinear parabolic equations with discontinuous diffusion coefficients
| Autor: | Alpár Richárd Mészáros, Dohyun Kwon |
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| Rok vydání: | 2019 |
| Předmět: |
General Mathematics
010102 general mathematics Mathematical analysis Degenerate energy levels Space (mathematics) 01 natural sciences 010101 applied mathematics Sobolev space Mathematics - Analysis of PDEs Free boundary problem FOS: Mathematics Differentiable function Uniqueness 0101 mathematics Diffusion (business) Probability measure Mathematics Analysis of PDEs (math.AP) |
| Zdroj: | Journal of the London Mathematical Society, 2021, Vol.104(2), pp.688-746 [Peer Reviewed Journal] |
| DOI: | 10.48550/arxiv.1907.11319 |
| Popis: | 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. Comment: 48 pages, 2 figures; to appear in J. Lond. Math. Soc |
| Databáze: | OpenAIRE |
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