Unstable Vortices, Sharp Non-Uniqueness with Forcing, and Global Smooth Solutions for the SQG Equation

Abstract

We prove that 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 that existence of global smooth solutions to the (unforced) \(\alpha \)-SQG equation that are neither rotating nor traveling.