A note on uniqueness of entropy solutions to degenerate parabolic equations in $\mathbb{R}^N$
| Autor: | Mohamed Maliki, Boris Andreianov |
|---|---|
| Přispěvatelé: | Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Université de Bourgogne (UB)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Equipe Modélisation, EDP et Analyse Numérique - FST Mohammédia, FST Mohammédia, DFG project 436 RUS 113/895/0-1 |
| Jazyk: | angličtina |
| Rok vydání: | 2010 |
| Předmět: |
Mathematics::Analysis of PDEs
01 natural sciences non-Lipschitz flux conservation law Initial value problem entropy solution [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] Uniqueness 0101 mathematics Primary 35K65 Secondary 35A05 Mathematical physics Mathematics Conservation law Applied Mathematics 010102 general mathematics Mathematical analysis Degenerate energy levels uniqueness Kato inequality infinite speed of propagation Parabolic partial differential equation 010101 applied mathematics Degenerate hyperbolic-parabolic equation Elliptic partial differential equation Gravitational singularity Hyperbolic partial differential equation Analysis |
| Zdroj: | Nonlinear Differential Equations and Applications Nonlinear Differential Equations and Applications, Springer Verlag, 2010, 17 (1), pp. 109-118. ⟨10.1007/s00030-009-0042-9⟩ |
| ISSN: | 1021-9722 1420-9004 |
| DOI: | 10.1007/s00030-009-0042-9⟩ |
| Popis: | The original publication is available at www.springerlink.com DOI: 10.1007/s00030-009-0042-9; International audience; We study the Cauchy problem in $\mathbb{R}^N$ for the parabolic equation $u_t+\text{div} F(u)=\Delta \varphi(u)$, which can degenerate into a hyperbolic equation for some intervals of values of $u$. In the context of conservation laws (the case $\varphi\equiv 0$), it is known that an entropy solution can be non-unique when $F'$ has singularities. We show the uniqueness of an entropy solution to the general parabolic problem for all $L^\infty$ initial datum, under the isotropic condition on the flux $F$ known for conservation laws. The only assumption on the diffusion term is that $\varphi$ is a non-decreasing continuous function. |
| Databáze: | OpenAIRE |
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