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Nonlinear Instability for the Surface Quasi-Geostrophic Equation in the Supercritical Regime

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Abstract

We consider the forced surface quasi-geostrophic equation with supercritical dissipation. We show that linear instability for steady state solutions leads to their nonlinear instability. When the dissipation is given by a fractional Laplacian, the nonlinear instability is expressed in terms of the scaling invariant norm, while we establish stronger instability claims in the setting of logarithmically supercritical dissipation. A key tool in treating the logarithmically supercritical setting is a global well-posedness result for the forced equation, which we prove by adapting and extending recent work related to nonlinear maximum principles. We believe that our proof of global well-posedness is of independent interest, to our knowledge giving the first large-data supercritical result with sharp regularity assumptions on the forcing term.

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Authors and Affiliations

  1. Department of Mathematics, Louisiana State University, Baton Rouge, LA, 70803-4918, USA

    Aynur Bulut

  2. Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI, 02912, USA

    Hongjie Dong

Authors
  1. Aynur Bulut
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  2. Hongjie Dong
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Correspondence to Aynur Bulut.

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Communicated by A. Ionescu.

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Bulut, A., Dong, H. Nonlinear Instability for the Surface Quasi-Geostrophic Equation in the Supercritical Regime. Commun. Math. Phys. 384, 1679–1707 (2021). https://doi.org/10.1007/s00220-021-04102-1

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  • DOI: https://doi.org/10.1007/s00220-021-04102-1

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