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 $\|\widehat{\theta^0}\|_{L^1}$ 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 $\lim_{t\rightarrow\infty}t^\delta\|\widehat{\theta}(t)\|_{L^1}=0$. Fourier analysis and standard techniques are used.