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Global in Time Classical Solutions to the 3D Quasi-Geostrophic System for Large Initial Data

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Abstract

In this paper, the authors show the existence of global in time classical solutions to the 3D quasi-geostrophic system with Ekman pumping for any smooth initial value (possibly large). This system couples an inviscid transport equation in \({\mathbb{R}^3_+}\) with an equation on the boundary satisfied by the trace. The proof combines the De Giorgi regularization effect on the boundary \({z=0}\)—similar to the so called surface quasi-geostrophic equation—with Beale–Kato–Majda techniques to propagate regularity for \({z > 0}\). A bootstrapping argument combining potential theory and Littlewood–Paley techniques is used to strengthen the regularization effect on the trace up to the Besov space B̊ 1∞,∞ .

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Authors and Affiliations

  1. Department of Mathematics, The University of Texas at Austin, Austin, TX, 78712, USA

    Matthew D. Novack & Alexis F. Vasseur

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  1. Matthew D. Novack
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  2. Alexis F. Vasseur
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Correspondence to Matthew D. Novack.

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Communicated by C. Mouhot

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Novack, M.D., Vasseur, A.F. Global in Time Classical Solutions to the 3D Quasi-Geostrophic System for Large Initial Data. Commun. Math. Phys. 358, 237–267 (2018). https://doi.org/10.1007/s00220-017-3049-9

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  • DOI: https://doi.org/10.1007/s00220-017-3049-9

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