Eulerian uniqueness of the $\alpha$-SQG patch problem

Xiaoyutao Luo

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arXiv · 2403.04219

Eulerian uniqueness of the $\alpha$-SQG patch problem

Xiaoyutao Luo

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Abstract

We consider the patch problem of the $\alpha$-SQG equation with $\alpha=0$ being the 2D Euler and $\alpha= \frac{1}{2}$ the SQG equations respectively. In the Eulerian setting, we prove the uniqueness of patch solutions of regularity $W^{2, \frac{1}{1-2\alpha} +} $ when $0<\alpha< \frac{1}{2}$ and $C^{1, 4\alpha+ }$ when $0<\alpha< \frac{1}{4} $. The proof is intrinsic to the modified Biot-Savart law and independent of the local existence of patch solutions.

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Xiaoyutao Luo. 2024-03-07. Eulerian uniqueness of the $\alpha$-SQG patch problem. https://arxiv.org/abs/2403.04219

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