Non-Decaying Solutions to the 2D Dissipative Quasi-Geostrophic Equations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913990936887296 |
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| author | Ambrose, David M. Aschoff, Ryan Cozzi, Elaine Kelliher, James P. |
| author_facet | Ambrose, David M. Aschoff, Ryan Cozzi, Elaine Kelliher, James P. |
| contents | We consider the surface quasi-geostrophic equation in two spatial dimensions, with subcritical diffusion (i.e. with fractional diffusion of order $2α$ for $α>\frac{1}{2}$.) We establish existence of solutions without assuming either decay at spatial infinity or spatial periodicity. One obstacle is that for $L^{\infty}$ data, the constitutive law may not be applicable, as Riesz transforms are unbounded. However, for $L^{\infty}$ initial data for which the constitutive law does converge, we demonstrate that there exists a unique solution locally in time, and that the constitutive law continues to hold at positive times. In the case that $α\in(\frac{1}{2},1]$ and that the initial data has some smoothness (specifically, if the data is in $C^{2}$), we demonstrate a maximum principle and show that this unique solution is actually classical and global in time. Then, a density argument allows us to show that mild solutions with only $L^{\infty}$ data are also global in time, and also possess this maximum principle. Finally, we introduce a related problem in which we replace the usual constitutive law for the surface quasi-geostrophic equation with a generalization of Sertfati type, and prove the same results for this relaxed model. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_10254 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-Decaying Solutions to the 2D Dissipative Quasi-Geostrophic Equations Ambrose, David M. Aschoff, Ryan Cozzi, Elaine Kelliher, James P. Analysis of PDEs We consider the surface quasi-geostrophic equation in two spatial dimensions, with subcritical diffusion (i.e. with fractional diffusion of order $2α$ for $α>\frac{1}{2}$.) We establish existence of solutions without assuming either decay at spatial infinity or spatial periodicity. One obstacle is that for $L^{\infty}$ data, the constitutive law may not be applicable, as Riesz transforms are unbounded. However, for $L^{\infty}$ initial data for which the constitutive law does converge, we demonstrate that there exists a unique solution locally in time, and that the constitutive law continues to hold at positive times. In the case that $α\in(\frac{1}{2},1]$ and that the initial data has some smoothness (specifically, if the data is in $C^{2}$), we demonstrate a maximum principle and show that this unique solution is actually classical and global in time. Then, a density argument allows us to show that mild solutions with only $L^{\infty}$ data are also global in time, and also possess this maximum principle. Finally, we introduce a related problem in which we replace the usual constitutive law for the surface quasi-geostrophic equation with a generalization of Sertfati type, and prove the same results for this relaxed model. |
| title | Non-Decaying Solutions to the 2D Dissipative Quasi-Geostrophic Equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2508.10254 |