Global Existence of Solutions to the 2D subcritical dissipative Quasi-Geostrophic equation and persistency of the initial regularity

In this paper, we prove that if the initial data $\theta_0$ and its Riesz transforms ($\mathcal{R}_1(\theta_0)$ and $\mathcal{R}_2(\theta_0)$) belong to the space $(\overline{S(\mathbb{R}^2))}^{B_{\infty}^{1-2\alpha ,\infty}}$, where $\alpha \in ]1/2,1[$, then the 2D Quasi-Geostrophic equation with dissipation $\alpha$ has a unique global in time solution $\theta$. Moreover, we show that if in addition $\theta_0 \in X$ for some functional space $X$ such as Lebesgue, Sobolev and Besov's spaces then the solution $\theta$ belongs to the space $C([0,+\infty [,X).$