Uniformly attracting limit sets for the critically dissipative SQG equation

Abstract

We consider the global attractor of the critical surface quasi-geostrophic (SQG) semigroup S ( t ) on the scale-invariant space () T H 12 . It was shown in [ 15 ] that this attractor is finite dimensional, and that it attracts uniformly bounded sets in () δ + T H 12 for any δ > 0 , leaving open the question of uniform attraction in () T H 12 . In this paper we prove the uniform attraction in () T H 12 , by combining ideas from the De Giorgi iteration and nonlinear maximum principles.