Asymptotic behavior of critical dissipative quasi-geostrophic equation in Fourier space

Abstract

In this paper we show the global existence for critical dissipative quasi-geostrophic equations if $\Vert \widehat{{\theta }^0}{\Vert }_{L^1}$ is small enough; among others we prove the analyticity of such a solution. If in addition the initial condition verifies $|D{|}^{-\delta }{\theta }^0\in L^1({\mathbb{R}}^2)$ with $0<\delta <1$, then the solution remains regular and ${\lim }_{t\to \infty }{t}^{\delta }\Vert \hat{\theta }(t){\Vert }_{L^1}=0$. Fourier analysis and standard techniques are used.