Asymptotic study of supercritical generalized SQG equations in critical Sobolev spaces

We study the long time behavior of regular solutions of the supercritical gSQG equations in the fully nonlinear regime. More precisely, under the assumption of small initial data in the critical Sobolev norm, we prove the existence of the unique global solution that satisfies the energy inequality (1.3) and for which the critical norm $\mathrm{\mid }\mathrm{\mid }\theta (t)\mathrm{\mid }{\mathrm{\mid }}_{H^{1+\beta -2\alpha }}$ decays to 0 as time goes to infinity.