Uniqueness of entropy solutions to fractional conservation laws with “fully infinite” speed of propagation

Abstract

Our goal is to study the uniqueness of bounded entropy solutions for a multidimensional conservation law including a non-Lipschitz convection term and a diffusion term of non-local porous medium type. The non-locality is given by a fractional power of the Laplace operator. For a wide class of nonlinearities, the L1-contraction principle is established, despite the fact that the “finite-infinite” speed of propagation (Alibaud (2007) [1]) cannot be exploited in our framework; existence is deduced with perturbation arguments. The method of proof, adapted from Andreianov and Maliki (2010) [9], requires a careful analysis of the action of the fractional laplacian on truncations of radial powers.