When Do Riemann Solutions Consist of Rarefactions, Jumps, and Constants?

TL;DR

This paper investigates the structure of Riemann solutions for hyperbolic conservation laws under minimal regularity assumptions, introducing new tools to classify continuous and discontinuous features.

Contribution

It introduces one-sided accumulation sets based on local essential images to analyze the structure of bounded solutions and characterizes when they are rarefactions, jumps, or constants.

Findings

Resonant solutions are rarefactions; non-resonant are constant.

All accumulation states of essential image discontinuities lie on a common Hugoniot locus.

Finite essential image discontinuities imply the solution is composed of finitely many waves and states.

Abstract

A solution of a Riemann problem for a strictly hyperbolic system of conservation laws is traditionally expected to consist of rarefaction waves, jump discontinuities, and constant states. In this paper, we investigate whether a Riemann solution has this structure when the solution is only assumed to be measurable and essentially bounded. To discriminate continuous and discontinuous features in an $L^\infty$ solution, we introduce one-sided accumulation sets based on local essential images. Supposing that throughout a bounded open interval a solution is continuous in the essential image (ess-im) sense, we prove that it is a rarefaction wave if it is resonant (the characteristic speed equals $x/t$), and otherwise it is constant. Although an ess-im discontinuity might not be a jump discontinuity, we show that all ess-im accumulation states lie on a common Hugoniot locus and have the same speed. Anomalies are possible if there are limit points of ess-im discontinuities, but if the set of ess-im discontinuities is finite, then an $L^\infty$ Riemann solution has bounded variation and is composed of finitely many rarefaction waves, jump discontinuities, and constant states.