  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 $C([0,+\infty [,X).$
