On the Forced Surface Quasi-Geostrophic Equation: Existence of Steady States and Sharp Relaxation Rates

We consider the asymptotic behavior of surface quasi-geostrophic equation, subject to a small external force. Under suitable assumptions on forcing, we first construct steady states and provide number useful posteriori estimates for them. Importantly, do so, only impose minimal cancellation conditions forcing function. Our main result is that all $$L^1\cap L^\infty $$ localized initial data produces global solutions forced SQG, which converge in $$L^p({\mathbf {R}}^2), 1<p\le 2$$ as time goes infinity. This establishes serve one point attracting set. Moreover, by employing method scaling variables, compute sharp relaxation rates, requiring slightly more data.