Eulerian uniqueness of the $α$-SQG patch problem
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916149707407360 |
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| author | Luo, Xiaoyutao |
| author_facet | Luo, Xiaoyutao |
| contents | We consider the patch problem of the $α$-SQG equation with $α=0$ being the 2D Euler and $α= \frac{1}{2}$ the SQG equations respectively. In the Eulerian setting, we prove the uniqueness of patch solutions of regularity $W^{2, \frac{1}{1-2α} +} $ when $0<α< \frac{1}{2}$ and $C^{1, 4α+ }$ when $0<α< \frac{1}{4} $. The proof is intrinsic to the modified Biot-Savart law and independent of the local existence of patch solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_04219 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Eulerian uniqueness of the $α$-SQG patch problem Luo, Xiaoyutao Analysis of PDEs We consider the patch problem of the $α$-SQG equation with $α=0$ being the 2D Euler and $α= \frac{1}{2}$ the SQG equations respectively. In the Eulerian setting, we prove the uniqueness of patch solutions of regularity $W^{2, \frac{1}{1-2α} +} $ when $0<α< \frac{1}{2}$ and $C^{1, 4α+ }$ when $0<α< \frac{1}{4} $. The proof is intrinsic to the modified Biot-Savart law and independent of the local existence of patch solutions. |
| title | Eulerian uniqueness of the $α$-SQG patch problem |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2403.04219 |