Explicit formulation for the Dirichlet problem for parabolic-hyperbolic conservation laws

Boris Andreianov (1, 2) , Mohamed Karimou Gazibo (2, 3)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
3 UAM - Abdou Moumouni University of Niamey = Université Abdou Moumouni de Niamey

We revisit the Cauchy-Dirichlet problem for degenerate parabolic scalar conservation laws. We suggest a new notion of strong entropy solution. It gives a straightforward explicit characterization of the boundary values of the solution and of the flux, and leads to a concise and natural uniqueness proof, compared to the one of the fundamental work [J. Carrillo, Arch. Ration. Mech. Anal., 1999]. Moreover, general dissipative boundary conditions can be studied in the same framework. The definition makes sense under the specific weak trace-regularity assumption. Despite the lack of evidence that generic solutions are trace-regular (especially in space dimension larger than one), the strong entropy formulation may be useful for modeling and numerical purposes.

Mots clés: Degenerate parabolic-hyperbolic equation; Dirichlet boundary condition; Strong entropy formulation; Weakly trace-regular solutions; Uniqueness proof

Domaines: Equations aux dérivées partielles [math.AP]

Boris Andreianov, Mohamed Karimou Gazibo. Explicit formulation for the Dirichlet problem for parabolic-hyperbolic conservation laws. 2015. ⟨hal-01152481v2⟩