Uniformly attracting limit sets for the critically dissipative SQG equation

TL;DR

This paper proves that the global attractor for the critical SQG equation in the scale-invariant space H^1(T^2) is uniformly attracting, using DeGiorgi iteration and nonlinear maximum principles.

Contribution

It establishes uniform attraction in H^1(T^2) for the critical SQG equation's global attractor, resolving an open question from prior research.

Findings

Global attractor is finite dimensional.

Attractor attracts uniformly bounded sets in H^{1+ ext{delta}}.

Proves uniform attraction in H^1(T^2).

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.