We prove that the critical surface quasi-geostrophic equation driven by a force f possesses a compact global attractor in $$L^2(\mathbb T^2)$$ L 2 ( T 2 ) provided $$f\in L^p(\mathbb T^2)$$ f ∈ L p ( T 2 ) for some $$p>2$$ p > 2 . First, the De Giorgi method is used to obtain uniform $$L^\infty $$ L ∞ estimates on viscosity solutions. Even though this does not provide a compact absorbing set, the existence of a compact global attractor follows from the continuity of solutions, which is obtained by estimating the energy flux using the Littlewood–Paley decomposition.
Journal of Mathematical Fluid Mechanics – Springer Journals
Published: May 30, 2017
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