Global structure of admissible BV solutions to piecewise genuinely nonlinear, strictly hyperbolic conservation laws in one space dimension

TL;DR

This paper characterizes the global structure of admissible BV solutions to strictly hyperbolic, piecewise genuinely nonlinear conservation laws, identifying a countable set of interaction points and Lipschitz curves where discontinuities occur.

Contribution

It extends previous structural results by providing a detailed description of the discontinuity set and the continuity regions of solutions, using wave-front tracking and convergence analysis.

Findings

Discontinuities are confined to a countable set of curves and points.

Solutions are continuous outside these curves and points.

The structural description applies to admissible BV solutions in the considered class.

Abstract

The paper gives an accurate description of the qualitative structure of an admissible BV solution to a strictly hyperbolic, piecewise genuinely nonlinear system of conservation laws. We prove that there are a countable set $\Theta$ which contains all interaction points and a family of countably many Lipschitz curves $\T$ such that outside $\T\cup \Theta$ $u$ is continuous, and along the curves in $\T$, u has left and right limit except for points in $\Theta$. This extends the corresponding structural result in \cite{BL,Liu1} for admissible solutions. The proof is based on approximate wave-front tracking solutions and a proper selection of discontinuity curves in the approximate solutions, which converge to curves covering the discontinuities in the exact solution $u$.