Steady and self-similar solutions of non-strictly hyperbolic systems of conservation laws

TL;DR

This paper analyzes steady, self-similar solutions to two-dimensional non-strictly hyperbolic conservation laws, showing they are BV functions and characterizing wave configurations, extending regularity and uniqueness results.

Contribution

It establishes that small perturbation solutions are BV functions and characterizes wave structures for non-strictly hyperbolic systems, extending prior regularity and uniqueness results.

Findings

Solutions are BV functions of bounded variation.

Wave configurations are explicitly characterized.

Regularity and uniqueness results are extended to certain Riemann problems.

Abstract

We consider solutions of two-dimensional $m \times m$ systems hyperbolic conservation laws that are constant in time and along rays starting at the origin. The solutions are assumed to be small $L^\infty$ perturbations of a constant state and entropy admissible, and the system is assumed to be non-strictly hyperbolic with eigenvalues of constant multiplicity. We show that such a solution, initially assumed bounded, must be a special function of bounded variation, and we determine the possible configuration of waves. As a corollary, we extend some regularity and uniqueness results for some one-dimensional Riemann problems.