From Navier-Stokes to discontinuous solutions of compressible Euler
The compressible Euler equations can develop shock discontinuities in finite time, as seen for instance in supersonic flows. A natural way to justify such singularities is to view Euler solutions as inviscid limits of Navier–Stokes flows with vanishing viscosity, but this limit is notoriously hard to control due to destabilizing viscous effects. After earlier results of Bianchini and Bressan for artificial viscosities, the corresponding problem for physical viscosities remained open. In this talk, I will review the classical mathematical framework for compressible fluid mechanics and explain a recent method of a-contraction with shifts, which allows us to describe the inviscid limit for the barotropic Euler equations and resolve the Bianchini–Bressan conjecture in this setting.