Shock formation in 1D conservation laws I: Inviscid structure

TL;DR

This paper investigates the stability, structure, and formation of shocks in one-dimensional hyperbolic conservation laws, providing detailed characterizations and expansions near singularities, with implications for vanishing viscosity limits.

Contribution

It offers a detailed analysis of shock formation stability, boundary characterization of classical development, and a fractional degree expansion for nondegenerate shocks in 1D conservation laws.

Findings

Shock formation is stable near simple waves.

Boundary of classical development is characterized near singularities.

Precise fractional degree expansion describes nondegenerate shocks.

Abstract

We study the stability and structure of shock formation in 1D hyperbolic conservation laws. We show that shock formation is stable near shocking simple waves: perturbations form a shock nearby in spacetime. We also characterize the boundary of the classical development in a spacetime neighborhood of the first time singularity. Finally, we describe the precise nature of nondegenerate shock formation through an expansion in homogeneous functions of fractional degree. We use these results in a companion paper to study the vanishing viscosity limit near shock formation.