Uniformly attracting limit sets for the critically dissipative SQG equation

Abstract

We consider the global attractor of the critical SQG semigroup $S(t)$ on the scale-invariant space $H^1({\mathbb{T}}^2)$. It was shown in~\cite{CTV13} that this attractor is finite dimensional, and that it attracts uniformly bounded sets in $H^{1+\delta }({\mathbb{T}}^2)$ for any $\delta >0$, leaving open the question of uniform attraction in $H^1({\mathbb{T}}^2)$. In this paper we prove the uniform attraction in $H^1({\mathbb{T}}^2)$, by combining ideas from DeGiorgi iteration and nonlinear maximum principles.

17 pages

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