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

Abstract

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

References in corpus (1)

• Well-posedness of the Cauchy problem for the fractional power dissipative equations