Global existence for the critical dissipative surface quasi-geostrophic equation

Abstract

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 ${\Lambda }^s({\dot{H}}_{uloc}^s({\mathbb{R}}^2))\cap L^{\infty }({\mathbb{R}}^2)$ 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 ($R\to \infty$, $\ep \rightarrow 0$) in $(SQG)_{R,\ep}$ and that the limit solution has the desired regularity.

References in corpus (1)

• Global well-posedness for the critical 2D dissipative quasi-geostrophic equation