# The Existence of a Global Attractor for the Forced Critical Surface Quasi-Geostrophic Equation in $$L^2$$ L 2

The Existence of a Global Attractor for the Forced Critical Surface Quasi-Geostrophic Equation in... 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Mathematical Fluid Mechanics Springer Journals

# The Existence of a Global Attractor for the Forced Critical Surface Quasi-Geostrophic Equation in $$L^2$$ L 2

, Volume 20 (1) – May 30, 2017
13 pages

/lp/springer_journal/the-existence-of-a-global-attractor-for-the-forced-critical-surface-ekj6lUTMUo
Publisher
Springer Journals
Copyright © 2017 by Springer International Publishing
Subject
Physics; Fluid- and Aerodynamics; Mathematical Methods in Physics; Classical and Continuum Physics
ISSN
1422-6928
eISSN
1422-6952
D.O.I.
10.1007/s00021-017-0324-7
Publisher site
See Article on Publisher Site

### Abstract

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

Journal of Mathematical Fluid MechanicsSpringer Journals

Published: May 30, 2017

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