Global existence for the critical dissipative surface quasi-geostrophic equation

In this article, we study the critical dissipative surface quasi-geostrophic equation (SQG) in $\mathbb{R}^2$. Motivated by the study of the homogeneous statistical solutions of this equation, we show that for any large initial data $\theta_{0}$ liying in the space $\Lambda^{s} (\dot H^{s}_{uloc}(\mathbb{R}^2)) \cap L^\infty(\mathbb{R}^2)$ the critical (SQG) has a global weak solution in time for all $1/2< s<1$. Our proof is based on an energy inequality verified by the truncated $(SQG)_{R,\ep}$ equation. By classical compactness arguments, we show that we are able to pass to the limit ($R \rightarrow \infty$, $\ep \rightarrow 0$) in $(SQG)_{R,\ep}$ and that the limit solution has the desired regularity.