### Abstract or Additional Information

Compressible Euler equations are a typical system of hyperbolic conservation laws,whose solution forms shock waves in general. It is well known that global BV solutions of system of hyperbolic conservation laws exist, when one considers small BV initial data. As a major breakthrough for system of hyperbolic conservation laws in 1990’s, by Bressan, Goatin, LeFloch and Lewicka, solutions have been proved to be unique among BV solutions verifying either the so-called Tame Oscillation Condition, or the Bounded Variation Condition on space-like curves.

In this talk, I will discuss the recent works with Krupa and Vasseur. In these works, for systems with two unknowns and the non-isentropic Euler equations with three unknowns, we established an L^2 stability theory using the method of relative entropy. As an application, we proved all BV solutions must statisfy the Bounded Variation Condition, hence showed the uniqueness of BV solution without any additional condition.

If time is permitted, I will briefly introduce the recent progress on the vanishing viscosity limit from Navier-Stokes equations to the BV solution of compressible Euler equations, which was missing from Bianchini-Bressan’s vanishing viscosity result. This is a collaboration work with Kang and Vasseur.