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Global Regularity for Several Incompressible Fluid Models with Partial Dissipation

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Abstract

This paper examines the global regularity problem on several 2D incompressible fluid models with partial dissipation. They are the surface quasi-geostrophic (SQG) equation, the 2D Euler equation and the 2D Boussinesq equations. These are well-known models in fluid mechanics and geophysics. The fundamental issue of whether or not they are globally well-posed has attracted enormous attention. The corresponding models with partial dissipation may arise in physical circumstances when the dissipation varies in different directions. We show that the SQG equation with either horizontal or vertical dissipation always has global solutions. This is in sharp contrast with the inviscid SQG equation for which the global regularity problem remains outstandingly open. Although the 2D Euler is globally well-posed for sufficiently smooth data, the associated equations with partial dissipation no longer conserve the vorticity and the global regularity is not trivial. We are able to prove the global regularity for two partially dissipated Euler equations. Several global bounds are also obtained for a partially dissipated Boussinesq system.

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Authors and Affiliations

  1. Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK, 74078, USA

    Jiahong Wu

  2. Laboratory of Mathematics and Complex Systems, Ministry of Education, School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Xiaojing Xu

  3. Department of Mathematics and Statistics, Jiangsu Normal University, 101 Shanghai Road, Xuzhou, 221116, Jiangsu, People’s Republic of China

    Zhuan Ye

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  1. Jiahong Wu
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  2. Xiaojing Xu
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  3. Zhuan Ye
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Correspondence to Zhuan Ye.

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Communicated by R. Shvydkoy

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Wu, J., Xu, X. & Ye, Z. Global Regularity for Several Incompressible Fluid Models with Partial Dissipation. J. Math. Fluid Mech. 19, 423–444 (2017). https://doi.org/10.1007/s00021-016-0291-4

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