From Navier-Stokes to discontinuous solutions of the compressible Euler

Abstract: The compressible Euler equation can lead to the emergence of shock discontinuities in finite time, notably observed behind supersonic planes. A very natural way to justify these singularities involves studying solutions as inviscid limits of Navier-Stokes solutions with evanescent viscosities. The mathematical study of this problem is however very difficult because of the destabilization effect of the viscosities.

Bianchini and Bressan proved the inviscid limit to small BV solutions using the so-called artificial viscosities in 2005. However, until very recently, achieving this limit with physical viscosities remained an open question.

In this presentation, we will  provide the basic ideas of  classical mathematical theories to compressible fluid mechanics and introduce the recent a-contraction with shifts method. This method is employed to describe the physical inviscid limit in the context of the barotropic Euler equation.