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hal-00378475, version 2

## A note on uniqueness of entropy solutions to degenerate parabolic equations in $\mathbb{R}^N$.

Boris Andreianov () 1, Mohamed Maliki () 2

NoDEA : Nonlinear Differential Equations and Applications 17, 1 (2010) 109-118

Abstract: 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.

• 1:  Laboratoire de Mathématiques (LM-Besançon)
• CNRS : UMR6623 – Université de Franche-Comté
• 2:  Equipe Modélisation, EDP et Analyse Numérique - FST Mohammédia
• FST Mohammédia
• Collaboration : DFG project 436 RUS 113/895/0-1
• Domain : Mathematics/Analysis of PDEs
• Keywords : Degenerate hyperbolic-parabolic equation – conservation law – non-Lipschitz flux – entropy solution – Kato inequality – infinite speed of propagation – uniqueness
• Comment : The original publication is available at www.springerlink.com DOI: 10.1007/s00030-009-0042-9
• Available versions :  v1 (2009-04-24) v2 (2009-09-05)

• hal-00378475, version 2
• oai:hal.archives-ouvertes.fr:hal-00378475
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• Submitted on: Saturday, 5 September 2009 19:33:35
• Updated on: Friday, 23 April 2010 11:54:02