Asymptotic study of a global solution of super-critical Quasi-Geostrophic equation

Abstract

In this paper, we study the super-critical Quasi-Geostrophic equation in Gevrey-Sobolev space. We prove the local existence of $(QG)$ for any large initial data and we give an exponential type of Blow-up to the solution. Moreover, we establish the existence global for a small initial data and we show that $\Vert \theta {\Vert }_{H_{a,{\alpha }^{-1}}^s}$ decays to zero as time goes to infinity. Fourier analysis and standard techniques are used.