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Long Time Dynamics of Forced Critical SQG

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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.

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  1. Department of Mathematics, Princeton University, Princeton, NJ, 08544, USA

    Peter Constantin, Andrei Tarfulea & Vlad Vicol

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  1. Peter Constantin
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Constantin, P., Tarfulea, A. & Vicol, V. Long Time Dynamics of Forced Critical SQG. Commun. Math. Phys. 335, 93–141 (2015). https://doi.org/10.1007/s00220-014-2129-3

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