The L2 theory for compressible Euler equations and vanishing viscosity limit from Navier-Stokes equations

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.

In this talk, I will first discuss the L2 stability result for systems with two unknowns and non-isentropic Euler equations with three unknowns. The main idea in these joint works with Krupa and Vasseur are to use the method of weighted relative entropy norm and modified front tracking sheme with shifts. As an application, we proved all BV solutions must statisfy the Bounded Variation Condition, which is a condition added by Bressan and etc to show uniqueness of solution. Hence we showed the uniqueness of BV solution.

Then I will briefly introduce the recent result on the vanishing viscosity limit from Navier-Stokes equations to the BV solution of compressible Euler equations. This is a famous open problem after Bressan-Bianchini’s seminal vanishing viscosity limit result for BV solutions of system of hyperbolic conservation laws. This is a joint work with Kang and Vasseur. And Kang will continue discussing it in his talk.