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Long-Time Behavior and Critical Limit of Subcritical SQG Equations in Scale-Invariant Sobolev Spaces

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Abstract

We consider the subcritical SQG equation in its natural scale-invariant Sobolev space and prove the existence of a global attractor of optimal regularity. The proof is based on a new energy estimate in Sobolev spaces to bootstrap the regularity to the optimal level, derived by means of nonlinear lower bounds on the fractional Laplacian. This estimate appears to be new in the literature and allows a sharp use of the subcritical nature of the \(L^\infty \) bounds for this problem. As a by-product, we obtain attractors for weak solutions as well. Moreover, we study the critical limit of the attractors and prove their stability and upper semicontinuity with respect to the strength of the diffusion.

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Acknowledgements

The author would like to thank Jacob Bedrossian, Alexey Cheskidov, Peter Constantin, Hao Jia and Vlad Vicol for helpful discussions. This work was supported in part by an AMS-Simons Travel Award.

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  1. Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    Michele Coti Zelati

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  1. Michele Coti Zelati
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Correspondence to Michele Coti Zelati.

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Communicated by Peter Constantin.

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Coti Zelati, M. Long-Time Behavior and Critical Limit of Subcritical SQG Equations in Scale-Invariant Sobolev Spaces. J Nonlinear Sci 28, 305–335 (2018). https://doi.org/10.1007/s00332-017-9409-y

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