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

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.