When Do Riemann Solutions Consist of Rarefactions, Jumps, and Constants?
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| Format: | Preprint |
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2026
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| author | Plohr, Bradley J. Schecter, Stephen Marchesin, Dan |
| author_facet | Plohr, Bradley J. Schecter, Stephen Marchesin, Dan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_21881 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | When Do Riemann Solutions Consist of Rarefactions, Jumps, and Constants? Plohr, Bradley J. Schecter, Stephen Marchesin, Dan Analysis of PDEs 35L65, 35L67 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. |
| title | When Do Riemann Solutions Consist of Rarefactions, Jumps, and Constants? |
| topic | Analysis of PDEs 35L65, 35L67 |
| url | https://arxiv.org/abs/2605.21881 |