## Uniqueness and Stability of Riemann Solutions with Large Oscillation in Gas Dynamics

#### Title: Uniqueness and Stability of Riemann Solutions with Large Oscillation in Gas Dynamics

Abstract
We prove the uniqueness of Riemann solutions in the class of entropy solutions in $L^\infty\cap BV_{loc}$ with arbitrarily large oscillation for the $3\X3$ system of Euler equations in gas dynamics. Our proof for solutions with {\it large} oscillation is based on a detailed analysis of the global behavior of shock curves in the phase space and the singularity of centered rarefaction waves near the center in the physical plane. The uniqueness of Riemann solutions yields their inviscid large-time stability under {\it arbitrarily large} $L^1\cap L^\infty\cap BV_{loc}$ perturbation of the Riemann initial data, as long as the corresponding solutions are in $L^\infty$ and have local bounded total variation satisfying a natural condition on its growth with time. No specific reference to any particular method for constructing the entropy solutions is made. Our uniqueness result for Riemann solutions can be easily extended to entropy solutions $U(x,t)$, piecewise Lipschitz in $x$, for any $t>0$, with arbitrarily large oscillation.
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