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 $B{V}_{\text{loc}}$. 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 [Arch. Ration. Mech. Anal., 222(1):343--391, 2016]), and introduce new ideas in this direction to allow for two shocks from different shock families to be controlled simultaneously. This paper shows $L^2$ stability for the Riemann problem for all time. Our results compare to Chen, Frid, and Li [Comm. Math. Phys., 228(2):201--217, 2002] and Chen and Li [J. Differential Equations, 202(2):332--353, 2004], 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 $L^2$ stability.

arXiv comment50 pages, 2 figures. This article uses the same hypotheses on the system and construction of the shift functions as arXiv:1904.09475

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Cited by in corpus (3)

• Uniqueness of a planar contact discontinuity for 3D compressible Euler system in a class of zero dissipation limits from Navier-Stokes-Fourier system
• Sharp a-contraction estimates for small extremal shocks
• Uniqueness & Weak-BV Stability in the Large for Isothermal Gas Dynamics