Liouville type theorem and kinetic formulation for 2x2 systems of conservation laws
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911073398947840 |
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| author | Ancona, Fabio Marconi, Elio Talamini, Luca |
| author_facet | Ancona, Fabio Marconi, Elio Talamini, Luca |
| contents | We study $\mathbf L^\infty$ entropy solutions to $2\times 2$ systems of conservation laws. We show that, if a uniformly convex entropy exists, these solutions satisfy a pair of kinetic equations (nonlocal in velocity), which are then shown to characterize all solutions with finite entropy production.
Next, we prove a Liouville-type theorem for genuinely nonlinear systems, which is the main result of the paper. This implies in particular that for every finite entropy solution, every point $(t,x) \in \mathbb R^+\times \mathbb R\setminus \br J$ is of vanishing mean oscillation, where $\br J \subset \mathbb R^+\times \mathbb R$ is a set of Hausdorff dimension at most 1. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_12840 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Liouville type theorem and kinetic formulation for 2x2 systems of conservation laws Ancona, Fabio Marconi, Elio Talamini, Luca Analysis of PDEs We study $\mathbf L^\infty$ entropy solutions to $2\times 2$ systems of conservation laws. We show that, if a uniformly convex entropy exists, these solutions satisfy a pair of kinetic equations (nonlocal in velocity), which are then shown to characterize all solutions with finite entropy production. Next, we prove a Liouville-type theorem for genuinely nonlinear systems, which is the main result of the paper. This implies in particular that for every finite entropy solution, every point $(t,x) \in \mathbb R^+\times \mathbb R\setminus \br J$ is of vanishing mean oscillation, where $\br J \subset \mathbb R^+\times \mathbb R$ is a set of Hausdorff dimension at most 1. |
| title | Liouville type theorem and kinetic formulation for 2x2 systems of conservation laws |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2502.12840 |