Global Existence, Hamiltonian Conservation and Vanishing Viscosity for the Surface Quasi-Geostrophic Equation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912563614187520 |
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| author | De Rosa, Luigi Latocca, Mickaël Park, Jaemin |
| author_facet | De Rosa, Luigi Latocca, Mickaël Park, Jaemin |
| contents | For any initial datum $θ_0\in L^{\frac{4}{3}}_x$ it is proved the existence of a global-in-time weak solution $θ\in L^\infty_t L^{\frac43}_x$ to the surface quasi-geostrophic equation whose Hamiltonian, i.e. the $\dot{H}^{-\frac{1}{2}}_x$ norm, is constant in time. The solution is obtained as a vanishing viscosity limit. Outside the classical strong compactness setting, the main idea is to propagate in time the non-concentration of the $L^{\frac{4}{3}}_x$ norm of the initial data, from which strong compactness in the Hamiltonian norm is deduced. General no anomalous dissipation results under minimal Onsager supercritical assumptions are also obtained. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_01268 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global Existence, Hamiltonian Conservation and Vanishing Viscosity for the Surface Quasi-Geostrophic Equation De Rosa, Luigi Latocca, Mickaël Park, Jaemin Analysis of PDEs For any initial datum $θ_0\in L^{\frac{4}{3}}_x$ it is proved the existence of a global-in-time weak solution $θ\in L^\infty_t L^{\frac43}_x$ to the surface quasi-geostrophic equation whose Hamiltonian, i.e. the $\dot{H}^{-\frac{1}{2}}_x$ norm, is constant in time. The solution is obtained as a vanishing viscosity limit. Outside the classical strong compactness setting, the main idea is to propagate in time the non-concentration of the $L^{\frac{4}{3}}_x$ norm of the initial data, from which strong compactness in the Hamiltonian norm is deduced. General no anomalous dissipation results under minimal Onsager supercritical assumptions are also obtained. |
| title | Global Existence, Hamiltonian Conservation and Vanishing Viscosity for the Surface Quasi-Geostrophic Equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2509.01268 |