Viscosity and the formation of shocks

Despite the small scales involved, the compressible Euler equations seem to be a good model even in the presence of shocks. Introducing viscosity is one way to resolve some of these small scale effects. In this talk, we examine the vanishing viscosity limit near the formation of a generic shock in one spatial dimension for a class of viscous conservation laws which includes compressible Navier Stokes. In a nutshell, we recover the inviscid (singular) solution in the limit, and we uncover universal structure in the viscous correctors. The proof involves matching approximate solutions constructed in regions where the viscosity is perturbative and where it is dominant. This is joint work with Sanchit Chaturvedi and Cole Graham.