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arXiv · 2605.21881

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

Bradley J. Plohr·Stephen Schecter·Dan Marchesin
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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.

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Bradley J. Plohr, Stephen Schecter, Dan Marchesin. 2026-05-21. When Do Riemann Solutions Consist of Rarefactions, Jumps, and Constants?. https://arxiv.org/abs/2605.21881

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