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The Existence of a Global Attractor for the Forced Critical Surface Quasi-Geostrophic Equation in \(L^2\)

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Abstract

We prove that the critical surface quasi-geostrophic equation driven by a force f possesses a compact global attractor in \(L^2(\mathbb T^2)\) provided \(f\in L^p(\mathbb T^2)\) for some \(p>2\). First, the De Giorgi method is used to obtain uniform \(L^\infty \) 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.

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois, Chicago, IL, 60607, USA

    Alexey Cheskidov & Mimi Dai

Authors
  1. Alexey Cheskidov
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  2. Mimi Dai
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Correspondence to Alexey Cheskidov.

Additional information

Communicated by R. Shvydkoy

Submission Date: October 26, 2015.

The work of Alexey Cheskidov was partially supported by NSF Grant DMS-1108864.

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Cheskidov, A., Dai, M. The Existence of a Global Attractor for the Forced Critical Surface Quasi-Geostrophic Equation in \(L^2\) . J. Math. Fluid Mech. 20, 213–225 (2018). https://doi.org/10.1007/s00021-017-0324-7

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  • DOI: https://doi.org/10.1007/s00021-017-0324-7

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