Finite time stability for the Riemann problem with extremal shocks for a large class of hyperbolic systems

Abstract

In this paper on hyperbolic systems of conservation laws in one space dimension, we give a complete picture of stability for all solutions to the Riemann problem which contain only extremal shocks. We study stability of the Riemann problem amongst a large class of solutions. We show stability among the family of solutions with shocks from any family. We assume solutions verify at least one entropy condition. We have no small data assumptions. The solutions we consider are bounded and satisfy a strong trace condition weaker than BVloc. We make only mild assumptions on the system. In particular, our work applies to gas dynamics, including the isentropic Euler system and the full Euler system for a polytropic gas. We use the theory of a-contraction (see Kang and Vasseur, 2016 [22]), and introduce new ideas in this direction to allow for two shocks from different shock families to be controlled simultaneously. This paper shows L2 stability for the Riemann problem for all time. Our results compare to Chen et al., 2002 [8] and Chen and Li, 2004 [9], which give uniqueness and long-time stability for perturbations of the Riemann problem – amongst a large class of solutions without smallness assumptions and which are locally BV. Although, these results lack global L2 stability.