Shock formation in 1D conservation laws I: Inviscid structure

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.