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We prove the uniqueness of Riemann solutions in the class of en- tropy solutions in L1\ BVloc for the 3 3 system of compressible Euler equations, under usual assumptions on the equation of state for the pressure which imply strict hyperbolicity of the system and genuine nonlinearity of the rst and third characteristic families. In particular, if the Riemann solutions consist of at most rarefaction waves and contact discontinuities, we show the global L2-stability of the Riemann solutions even in the class of entropy solu- tions in L1 with arbitrarily large oscillation for the 3 3 system. We apply our framework established earlier to show that the uniqueness of Riemann so- lutions implies their inviscid asymptotic stability under L1 perturbation of the Riemann initial data, as long as the corresponding solutions are in L1 and have local bounded total variation satisfying a natural condition on its growth with time. No specic reference to any particular method for constructing the entropy solutions is made. Our uniqueness result for Riemann solutions can easily be extended to entropy solutions U(x;t), piecewise Lipschitz in x ,f or any t> 0.

Uniqueness and asymptotic stability of Riemann solutions for the compressible Euler equations