Global existence for the critical dissipative surface quasi-geostrophic equation

TL;DR

This paper proves the global existence of weak solutions for the critical dissipative surface quasi-geostrophic equation in two dimensions for large initial data within certain function spaces.

Contribution

It establishes the existence of global weak solutions for the critical SQG with large initial data in specific function spaces, using energy inequalities and compactness methods.

Findings

Global weak solutions exist for all initial data in specified spaces.

The proof employs energy inequalities and compactness arguments.

Solutions have the desired regularity after passing to the limit.

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 $\Lambda^{s} (\dot H^{s}_{uloc}(\mathbb{R}^2)) \cap L^\infty(\mathbb{R}^2)$ 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.