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Long Time Dynamics of Forced Critical SQG

Author(s): Constantin, Peter; Tarfulea, Andrei; Vicol, Vlad C.

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Abstract: We prove the existence of a compact global attractor for the dynamics of the forced critical surface quasi-geostrophic equation (SQG) and prove that it has finite fractal (box-counting) dimension. In order to do so we give a new proof of global regularity for critical SQG. The main ingredient is the nonlinear maximum principle in the form of a nonlinear lower bound on the fractional Laplacian, which is used to bootstrap the regularity directly from L ∞ to C α , without the use of De Giorgi techniques. We prove that for large time, the norm of the solution measured in a sufficiently strong topology becomes bounded with bounds that depend solely on norms of the force, which is assumed to belong merely to L ∞ ∩ H 1 . Using the fact that the solution is bounded independently of the initial data after a transient time, in spaces conferring enough regularity, we prove the existence of a compact absorbing set for the dynamics in H 1 , obtain the compactness of the linearization and the continuous differentiability of the solution map. We then prove exponential decay of high yet finite dimensional volume elements in H 1 along solution trajectories, and use this property to bound the dimension of the global attractor.
Publication Date: Apr-2015
Electronic Publication Date: 9-Aug-2014
Citation: Constantin, Peter, Tarfulea, Andrei, Vicol, Vlad. (2015). Long Time Dynamics of Forced Critical SQG. COMMUNICATIONS IN MATHEMATICAL PHYSICS, 335 (93 - 141. doi:10.1007/s00220-014-2129-3
DOI: doi:10.1007/s00220-014-2129-3
ISSN: 0010-3616
EISSN: 1432-0916
Pages: 93 - 141
Type of Material: Journal Article
Version: Author's manuscript

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