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Journal Articles Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik Year : 2013

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

Boris Andreianov
Mohamed Karimou Gazibo
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#### 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.

#### Domains

Mathematics [math] Analysis of PDEs [math.AP]

### Dates and versions

hal-00697593 , version 1 (15-05-2012)
hal-00697593 , version 2 (04-10-2012)

### Licence

Attribution - NonCommercial - CC BY 4.0

### Identifiers

• HAL Id : hal-00697593 , version 2
• DOI :

### Cite

Boris Andreianov, Mohamed Karimou Gazibo. Entropy formulation of degenerate parabolic equation with zero-flux boundary condition. Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 2013, 164 (5), pp. 1471-1491. ⟨10.1007/s00033-012-0297-6⟩. ⟨hal-00697593v2⟩

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