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

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.