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Vanishing Diffusion Limits and Long Time Behaviour of a Class of Forced Active Scalar Equations

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Abstract

We investigate the properties of an abstract family of advection diffusion equations in the context of the fractional Laplacian. Two independent diffusion parameters enter the system, one via the constitutive law for the drift velocity and one as the prefactor of the fractional Laplacian. We obtain existence and convergence results in certain parameter regimes and limits. We study the long time behaviour of solutions to the general problem and prove the existence of a unique global attractor. We apply the results to two particular active scalar equations arising in geophysical fluid dynamics, namely the surface quasigeostrophic equation and the magnetogeostrophic equation.

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Notes

  1. We point out that most of the results given in our work hold for \(d{\,\geqq \,}2\).

  2. Such a mean zero assumption is common in many physical models which include SQG equation and MG equation; see [9] and [17], for example.

  3. \(L^\infty \) is the critical Lebesgue space with respect to the natural scaling for both the critically diffusive SQG and MG\(^0\) equations.

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Acknowledgements

We would like to thank the reviewers for their valuable comments and suggestions which helped to improve the manuscript. S. Friedlander is supported by NSF DMS-1613135 and A. Suen is supported by the Hong Kong General Research Fund (GRF) Grant Project number 18300720.

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Authors and Affiliations

  1. Department of Mathematics, University of Southern California, Los Angeles, USA

    Susan Friedlander

  2. Department of Mathematics and Information Technology, The Education University of Hong Kong, Hong Kong, China

    Anthony Suen

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  1. Susan Friedlander
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Correspondence to Anthony Suen.

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Communicated by V. Vicol.

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Friedlander, S., Suen, A. Vanishing Diffusion Limits and Long Time Behaviour of a Class of Forced Active Scalar Equations. Arch Rational Mech Anal 240, 1431–1485 (2021). https://doi.org/10.1007/s00205-021-01638-3

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