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

Boris Andreianov (1) , Mohamed Maliki (2)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
2 Equipe Modélisation, EDP et Analyse Numérique - FST Mohammédia

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.

Mots clés: Degenerate hyperbolic-parabolic equation; conservation law; non-Lipschitz flux; entropy solution; Kato inequality; infinite speed of propagation; uniqueness

Domaines: Equations aux dérivées partielles [math.AP]

Boris Andreianov, Mohamed Maliki. A note on uniqueness of entropy solutions to degenerate parabolic equations in $\mathbb{R}^N$.. Nonlinear Differential Equations and Applications, 2010, 17 (1), pp. 109-118. ⟨10.1007/s00030-009-0042-9⟩. ⟨hal-00378475v2⟩