Formation and construction of large variational shock waves for 1-D $n\times n$ quasilinear hyperbolic conservation systems

TL;DR

This paper investigates the formation and construction of large variational shock waves in 1-D quasilinear hyperbolic conservation systems, extending previous blowup analysis to include shock wave development from singularities.

Contribution

It introduces a method to construct large variational shock waves from blowup points in 1-D hyperbolic systems with genuinely nonlinear eigenvalues, building on prior blowup mechanism studies.

Findings

Established the formation of large variational shock waves from blowup points.

Provided a detailed construction method for shock waves in the specified systems.

Extended understanding of singularity behavior in hyperbolic conservation laws.

Abstract

In the paper [Li Jun, Xu Gang, Yin Huicheng, On the blowup mechanism of smooth solutions to 1D quasilinear strictly hyperbolic systems with large variational initial data, Nonlinearity 38 (2025), No.2, 025016], for the 1-D $n\times n$ ($n\geqslant 3$) strictly hyperbolic system $\partial_tv+F(v)\partial_xv=0$ with some classes of large variational initial data $v(x, 0)$, the geometric blowup mechanism and the detailed singularity behaviours of $\partial_{x,t}v$ near the blowup point are studied when the $n\times n$ matrix $F(v)$ admits at least one genuinely nonlinear eigenvalue. In this paper, we focus on the formation and construction of a large variational shock wave from the blowup point for 1-D $n\times n$ quasilinear hyperbolic conservation law system $\partial_tu+\partial_xf(u)=0$ when some smooth simple wave solution is generic non-degenerate before the formation of singularity and the corresponding eigenvalue is genuinely nonlinear.