Shock formation in 1D conservation laws II: Vanishing viscosity

TL;DR

This paper investigates how small viscosity influences shock formation in 1D hyperbolic conservation laws, providing precise convergence rates and universal viscous behavior near singularities using asymptotic analysis.

Contribution

It introduces a novel approximation scheme to analyze viscous effects on shocks, applicable to complex systems like Navier–Stokes with degenerate viscosity.

Findings

Established sharp convergence rates up to shock formation

Identified universal viscous behavior near singularities

Developed a decoupling-based approximation scheme

Abstract

We study the effects of weak viscosity on shock formation in 1D hyperbolic conservation laws. Given an inviscid solution that forms a nondegenerate shock, we add a small viscous regularization and study the limit as the viscosity vanishes. Using a matched asymptotic expansion, we determine the sharp rate of convergence in strong norms up to the time of inviscid shock formation, and we identify universal viscous behavior near the first singularity. To treat the complex interactions between multiple characteristics and the viscosity, we develop an approximation scheme that exploits a certain decoupling between shocking and nonshocking characteristics. Our analysis makes minimal assumptions on the equation, and in particular applies to the compressible Navier--Stokes equations with degenerate physical viscosity.