Entropy conditions for scalar conservation laws with discontinuous flux revisited

TL;DR

This paper introduces new entropy conditions for multidimensional scalar conservation laws with discontinuous flux, improving the characterization of solutions and their limits, and extending previous one-dimensional results to more complex, inhomogeneous cases.

Contribution

It generalizes entropy admissibility conditions to higher dimensions without requiring the crossing condition, and establishes uniqueness and existence tools for solutions with discontinuous flux.

Findings

Proved uniqueness of solutions under the new entropy conditions.

Provided a framework for the vanishing viscosity approximation in multidimensional cases.

Extended the theory to inhomogeneous flux with multiple discontinuity hypersurfaces.

Abstract

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. \smallskip 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.