Entropy conditions for scalar conservation laws with discontinuous flux revisited

  • Darko Mitrović

    University of Montenegro, Faculty of Natural Sciences and Mathematics, Cetinjski put bb, 81000 Podgorica, Montenegro
  • Boris Andreianov

    Laboratoire de Mathématiques CNRS UMR 6623, Université de Franche-Comté, 25030 Besançon Cedex, France, Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany


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 Karlsen, Risebro, and Towers (2003) [31] 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.

Cite this article

Darko Mitrović, Boris Andreianov, Entropy conditions for scalar conservation laws with discontinuous flux revisited. Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015), no. 6, pp. 1307–1335

DOI 10.1016/J.ANIHPC.2014.08.002