Unstable vortices, sharp non-uniqueness with forcing, and global smooth solutions for the SQG equation
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913691124891648 |
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| author | Castro, Ángel Faraco, Daniel Mengual, Francisco Solera, Marcos |
| author_facet | Castro, Ángel Faraco, Daniel Mengual, Francisco Solera, Marcos |
| contents | We prove non-uniqueness of weak solutions to the forced $α$-SQG equation with Sobolev regularity $W^{s,p}$ in the supercritical regime $s < α+ \frac{2}{p}$, covering the 2D Euler equation ($α= 0$), the Surface Quasi-Geostrophic equation ($α= 1$), and the intermediate cases. A key step is the construction of smooth, compactly supported vortices that exhibit non-linear instability. As a by-product, we show existence of global smooth solutions to the (unforced) $α$-SQG equation that are neither rotating nor traveling. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_10274 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Unstable vortices, sharp non-uniqueness with forcing, and global smooth solutions for the SQG equation Castro, Ángel Faraco, Daniel Mengual, Francisco Solera, Marcos Analysis of PDEs We prove non-uniqueness of weak solutions to the forced $α$-SQG equation with Sobolev regularity $W^{s,p}$ in the supercritical regime $s < α+ \frac{2}{p}$, covering the 2D Euler equation ($α= 0$), the Surface Quasi-Geostrophic equation ($α= 1$), and the intermediate cases. A key step is the construction of smooth, compactly supported vortices that exhibit non-linear instability. As a by-product, we show existence of global smooth solutions to the (unforced) $α$-SQG equation that are neither rotating nor traveling. |
| title | Unstable vortices, sharp non-uniqueness with forcing, and global smooth solutions for the SQG equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2502.10274 |