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In this paper on hyperbolic systems of conservation laws in one spacedimension, we give a complete picture of stability for all solutions to theRiemann problem which contain only extremal shocks. We study stability of theRiemann problem amongst a large class of solutions. We show stability among thefamily of solutions with shocks from any family. We assume solutions verify atleast one entropy condition. We have no small data assumptions. The solutionswe consider are bounded and satisfy a strong trace condition weaker than$BV_{\text{loc}}$. We make only mild assumptions on the system. In particular,our work applies to gas dynamics, including the isentropic Euler system and thefull Euler system for a polytropic gas. We use the theory of a-contraction (seeKang and Vasseur [Arch. Ration. Mech. Anal., 222(1):343--391, 2016]), andintroduce new ideas in this direction to allow for two shocks from differentshock 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 andlong-time stability for perturbations of the Riemann problem -- amongst a largeclass of solutions without smallness assumptions and which are locally $BV$.Although, these results lack global $L^2$ stability.

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