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 $$\alpha $$ α -SQG equation with Sobolev regularity $$W^{s,p}$$ W s , p in the supercritical regime $$s s α + 2 p , covering the 2D Euler equation ( $$\alpha = 0$$ α = 0 ), the Surface Quasi-Geostrophic equation ( $$\alpha = 1$$ α = 1 ), and the intermediate cases. A key step is the construction of smooth, compactly supported vortices that exhibit non-linear instability. As a by-product, we show that existence of global smooth solutions to the (unforced) $$\alpha $$ α -SQG equation that are neither rotating nor traveling.