Quantitative inviscid limits and universal shock development in scalar conservation laws

We explore one facet of an old problem: the approximation of hyperbolic conservation laws by viscous counterparts. While qualitative convergence results are well-known, quantitative rates for the inviscid limit are less common. In this talk, we consider the simplest case: a one-dimensional scalar strictly-convex conservation law started from "generic" smooth initial data. Using a matched asymptotic expansion, we quantitatively control the inviscid limit up to the time of first shock. We conclude that the inviscid limit has a universal character near the first shock which is independent of the flux and initial data (up to scaling). This is joint work with Sanchit Chaturvedi.