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Journal of Mathematical Fluid Mechanics

, Volume 19, Issue 3, pp 423–444 | Cite as

Global Regularity for Several Incompressible Fluid Models with Partial Dissipation

  • Jiahong Wu
  • Xiaojing Xu
  • Zhuan Ye
Article

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.

Keywords

Euler equation surface quasi-geostrophic equation Boussinesq equations partial dissipation global regularity 

Mathematics Subject Classification

35A05 35B45 35B65 76D03 76D09 

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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of MathematicsOklahoma State UniversityStillwaterUSA
  2. 2.Laboratory of Mathematics and Complex Systems, Ministry of EducationSchool of Mathematical Sciences, Beijing Normal UniversityBeijingPeople’s Republic of China
  3. 3.Department of Mathematics and StatisticsJiangsu Normal UniversityXuzhouPeople’s Republic of China

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