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 $1\mathrm{/}2<s<1$. Our proof is based on an energy inequality verified by the truncated $(SQG{)}_{R,\text{ep}}$ equation. By classical compactness arguments, we show that we are able to pass to the limit ($R\to \mathrm{\infty }$, $\ep \rightarrow 0$)in$(SQG){R,\ep}$ and that the limit solution has the desired regularity.