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hal-00697593, version 1

## Entropy formulation of degenerate parabolic equation with zero-flux boundary condition

Boris Andreianov () 1, Mohamed Karimou Gazibo () 1

(2012-05-15)

Abstract: We consider the general degenerate hyperbolic-parabolic equation: $$\label{E}\tag{E} u_t+\div f(u)-\Delta\phi(u)=0 \mbox{ in } Q = (0,T)\times\Omega,\;\;\;\; T>0,\;\;\;\Omega\subset\mathbb R^N ;$$ with initial condition and the zero flux boundary condition. Here $\phi$ is a continuous non decreasing function. Following [B\"{u}rger, Frid and Karlsen, J. Math. Anal. Appl, 2007], we assume that $f$ is compactly supported (this is the case in several applications) and we define an appropriate notion of entropy solution. Using vanishing viscosity approximation, we prove existence of entropy solution for any space dimension $N\geq 1$ under a partial genuine nonlinearity assumption on $f$. Uniqueness is shown for the case $N=1$, using the idea of [Andreianov and Bouhsiss, J. Evol. Equ., 2004], nonlinear semigroup theory and a specific regularity result for one dimension.

• 1:  Laboratoire de Mathématiques (LM-Besançon)
• CNRS : UMR6623 – Université de Franche-Comté
• Domain : Mathematics/Analysis of PDEs
• Keywords : Degenerate hyperbolic-parabolic equation – Zero-flux boundary condition – Vanishing viscosity approximation – Nonlinear semigroup theory – Entropy solution – Integral solution
• Comment : 19 pages
• Available versions :  v1 (2012-05-15) v2 (2012-10-04)

• hal-00697593, version 1
• oai:hal.archives-ouvertes.fr:hal-00697593
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• Submitted on: Tuesday, 15 May 2012 16:45:49
• Updated on: Thursday, 4 October 2012 20:55:14