Existence of global solutions to the 2-D subcritical dissipative quasi-geostrophic equation and persistency of the initial regularity

Abstract

In this article, we prove that if the initial data $heta_0$ and its Riesz transforms ($mathcal{R}_1(heta_0)$ and $mathcal{R}_2(heta_0)$) belong to the space $$ (overline{S(mathbb{R}^2))} ^{B_{infty }^{1-2alpha ,infty }}, quad 1/2<alpha <1, $$ then the 2-D Quasi-Geostrophic equation with dissipation $alpha$ has a unique global in time solution $heta$. Moreover, we show that if in addition $heta_0 in X$ for some functional space $X$ such as Lebesgue, Sobolev and Besov's spaces then the solution $heta$ belongs to the space $C([0,+infty [,X)$.

Keywords

• Quasi-geostrophic equation
• Besov Spaces