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arXiv · 2502.10274

Unstable vortices, sharp non-uniqueness with forcing, and global smooth solutions for the SQG equation

Ángel Castro·Daniel Faraco·Francisco Mengual·Marcos Solera
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Abstract

We prove non-uniqueness of weak solutions to the forced $\alpha$-SQG equation with Sobolev regularity $W^{s,p}$ in the supercritical regime $s < \alpha + \frac{2}{p}$, covering the 2D Euler equation ($\alpha = 0$), the Surface Quasi-Geostrophic equation ($\alpha = 1$), 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) $\alpha$-SQG equation that are neither rotating nor traveling.

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Ángel Castro, Daniel Faraco, Francisco Mengual, Marcos Solera. 2025-02-14. Unstable vortices, sharp non-uniqueness with forcing, and global smooth solutions for the SQG equation. https://arxiv.org/abs/2502.10274

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