临界耗散SQG方程的一致吸引极限集
Uniformly attracting limit sets for the critically dissipative SQG equation
摘要：
研究了尺度不变空间$H^1(\mathbb{T}^2)$上临界SQG半群$S(t)$的全局吸引子.在~ \cite{CTV13}中证明了这个吸引子是有限维的，并且对于任何$δ>0$，它吸引$H^{1+δ}(\mathbb{T}^2)$中的一致有界集，留下了$H^1(\mathbb{T}^2)$中的一致吸引问题。本文结合DeGiorgi迭代法和非线性最大值原理的思想，证明了$H^1(\mathbb{T}^2)$中的一致吸引.
英文摘要：
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 $H^{1+δ}(\mathbb{T}^2)$ for any $δ>0$, 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.
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