Stability of small BV solutions to compressible Euler system in the class of inviscid limits from Navier-Stokes

The convex integration shows that the compressible Euler system in multi-D is ill-posed in the class of entropy solutions, especially, non-uniqueness of entropy solutions containing a shock wave.

Recently, for the 1D isentropic case, we showed the uniqueness and stability of entropy solutions of small BV in the class of vanishing physical viscosity limits, that is, inviscid limits from the associated Navier-Stokes system.

These results use the so-called 'a-contraction with shifts' method for handling the uniform stability of a viscous shock.

In this talk, I will explain the key ideas of the method of a-contraction with shifts for a single viscous shock, and extension of the method to more general situations for two shocks and small BV solutions. Also, I will briefly mention its application to the study on long-time behavior of Navier-Stokes flows perturbed from Riemann data.

The results of this talk are based on the joint works with G. Chen, A. Vasseur, Y. Wang and other collaborators in Korea.