Global weak solutions of a Hamiltonian regularised Burgers equation
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866929263838494720 |
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| author | Guelmame, Billel Junca, Stéphane Clamond, Didier Pego, Robert L. |
| author_facet | Guelmame, Billel Junca, Stéphane Clamond, Didier Pego, Robert L. |
| contents | A nondispersive, conservative regularisation of the inviscid Burgers equation is proposed and studied. Inspired by a related regularisation of the shallow water system recently introduced by Clamond and Dutykh, the new regularisation provides a family of Galilean-invariant interpolants between the inviscid Burgers equation and the Hunter-Saxton equation. It admits weakly singular regularised shocks and cusped traveling-wave weak solutions. The breakdown of local smooth solutions is demonstrated, and the existence of two types of global weak solutions, conserving or dissipating an $H^1$ energy, is established. Dissipative solutions satisfy an Oleinik inequality like entropy solutions of the inviscid Burgers equation. As the regularisation scale parameter $\ell$ tends to $0$ or $\infty$, limits of dissipative solutions are shown to satisfy the inviscid Burgers or Hunter-Saxton equation respectively, forced by an unknown remaining term. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_15545 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global weak solutions of a Hamiltonian regularised Burgers equation Guelmame, Billel Junca, Stéphane Clamond, Didier Pego, Robert L. Analysis of PDEs A nondispersive, conservative regularisation of the inviscid Burgers equation is proposed and studied. Inspired by a related regularisation of the shallow water system recently introduced by Clamond and Dutykh, the new regularisation provides a family of Galilean-invariant interpolants between the inviscid Burgers equation and the Hunter-Saxton equation. It admits weakly singular regularised shocks and cusped traveling-wave weak solutions. The breakdown of local smooth solutions is demonstrated, and the existence of two types of global weak solutions, conserving or dissipating an $H^1$ energy, is established. Dissipative solutions satisfy an Oleinik inequality like entropy solutions of the inviscid Burgers equation. As the regularisation scale parameter $\ell$ tends to $0$ or $\infty$, limits of dissipative solutions are shown to satisfy the inviscid Burgers or Hunter-Saxton equation respectively, forced by an unknown remaining term. |
| title | Global weak solutions of a Hamiltonian regularised Burgers equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2402.15545 |