Entropy conditions for scalar conservation laws with discontinuous flux revisited

Boris Andreianov (1, 2) , Darko Mitrović (3)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
2 Institut für Mathematik [Berlin]
3 Faculty of Natural Studies and Mathematics

We propose new entropy admissibility conditions for multidimensional hyperbolic scalar conservation laws with discontinuous flux which generalize one-dimensional Karlsen-Risebro-Towers entropy conditions. These new conditions are designed, in particular, in order to characterize the limit of vanishing viscosity approximations. On the one hand, they comply quite naturally with a certain class of physical and numerical modeling assumptions; on the other hand, their mathematical assessment turns out to be intricate. The generalization we propose is not only with respect to the space dimension, but mainly in the sense that the ''crossing condition'' of [K.H. Karlsen, N.H. Risebro, J. Towers, Skr.\,K.\,Nor.\,Vid.\,Selsk. (2003)] is not mandatory for proving uniqueness with the new definition. We prove uniqueness of solutions and give tools to justify their existence via the vanishing viscosity method, for the multi-dimensional spatially inhomogeneous case with a finite number of Lipschitz regular hypersurfaces of discontinuity for the flux function.

Mots clés: inhomogeneous scalar conservation law; discontinuous flux; change of variables; entropy solution; vanishing viscosity approximation; well-posedness; crossing condition

Domaines: Equations aux dérivées partielles [math.AP]

Boris Andreianov, Darko Mitrović. Entropy conditions for scalar conservation laws with discontinuous flux revisited. Ann. IHP Analyse Nonlinéaire, 2015, ⟨10.1016/j.anihpc.2014.08.002⟩. ⟨hal-00978482v2⟩