Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - Université de Rouen Normandie

We will start by recalling the main facts about the theory of entropy solutions to hyperbolic conservation laws. In particular, Kruzkov entropies single out the "good" solution among all weak solutions to a general scalar balance law. This is coherent with the selection based on any other convex entropy E. However we will explain the limitations one encounters when limiting oneself to this framework. In particular when dealing with the analysis of traffic flow. This motivates the introduction of a general entropy production operator and the analysis of its functional properties. Topological considerations constrain the framework to the projective tensor product of C^1 spaces for fluxes, entropies and entropy fluxes. Surprisingly, complex valued entropies and Fourier transform techniques provide families of entropies, alternative to the Kruzkov one, still able to characterize this operator. As a byproduct, this general approach also shows that the entropy production furnishes a faithful representation of any essentially bounded function. This talk is based on joint works with Rinaldo M. Colombo.