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 α α -SQG equation with Sobolev regularity Ws,p W s , p in the supercritical regime s < α + 2/p s < α + 2 p , covering the 2D Euler equation ( α = 0 α = 0 ), the Surface Quasi-Geostrophic equation ( α = 1 α = 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) α α -SQG equation that are neither rotating nor traveling.