Formation and construction of a shock wave for one dimensional $n\times n$ strictly hyperbolic conservation laws with small smooth initial data

TL;DR

This paper analyzes shock wave formation in 1-D strictly hyperbolic conservation laws with small initial data, describing blowup behavior and constructing weak entropy shocks, with applications to various physical systems.

Contribution

It provides a detailed description of blowup rates, cusp singularity structures, and constructs shock waves for hyperbolic systems, extending understanding of singularity formation and shock development.

Findings

Precise space-time blowup rate near the singularity.

Description of cusp singularity structure of characteristic envelope.

Construction of weak entropy shock starting from blowup point.

Abstract

Under the genuinely nonlinear assumption for 1-D $n\times n$ strictly hyperbolic conservation laws, we investigate the geometric blowup of smooth solutions and the development of singularities when the small initial data fulfill the generic nondegenerate condition. At first, near the unique blowup point we give a precise description on the space-time blowup rate of the smooth solution and meanwhile derive the cusp singularity structure of characteristic envelope. These results are established through extending the smooth solution of the completely nonlinear blowup system across the blowup time. Subsequently, by utilizing a new form on the resulting 1-D strictly hyperbolic system with $(n-1)$ good components and one bad component, together with the choice of an efficient iterative scheme and some involved analyses, a weak entropy shock wave starting from the blowup point is constructed. As a byproduct, our result can be applied to the shock formation and construction for the 2-D supersonic steady compressible full Euler equations ($4\times 4$ system), 1-D MHD equations ($5\times 5$ system), 1-D elastic wave equations ($6\times 6$ system) and 1-D full ideal compressible MHD equations ($7\times 7$ system).