Uniqueness for the Dafermos regularization viscous wave fan profiles for Riemann solutions of scalar hyperbolic conservation laws

Abstract

We prove the uniqueness of solutions to the Dafermos regularization viscous wave fan profiles for Riemann solutions of scalar hyperbolic conservation laws. We emphasize that our results are not restricted to the small self-similar viscosity regime. We rely on suitable adaptations of Serrin's sweeping principle and the sliding method from the qualitative theory of semilinear elliptic PDEs. In order to illustrate the delicacy of our result, we prove the existence of an unbounded solution in the case of Burgers equation. Lastly, we can combine aspects of these results in order to give a precise description of the Dafermos regularization of rarefaction waves of Burgers equation.