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On the Global Well-Posedness of 3-D Boussinesq System with Variable Viscosity

Abstract

In this paper, the authors first consider the global well-posedness of 3-D Boussinesq system, which has variable kinematic viscosity yet without thermal conductivity and buoyancy force, provided that the viscosity coefficient is sufficiently close to some positive constant in L and the initial velocity is small enough in \(\dot{B}_{3,1}^0(\mathbb{R}^3)\). With some thermal conductivity in the temperature equation and with linear buoyancy force θe3 on the velocity equation in the Boussinesq system, the authors also prove the global well-posedness of such system with initial temperature and initial velocity being sufficiently small in L1(ℝ3) and \(\dot{B}_{3,1}^0(\mathbb{R}^3)\) respectively.

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Acknowledgement

Part of this work was done when we were visiting Morningside Center of Mathematics, CAS, in the summer of 2013. We appreciate the hospitality and the financial support from the Center.

Author information

Correspondence to Hammadi Abidi or Ping Zhang.

Additional information

Dedicated to Professor Andrew J. Majda for the 70th birthday

This work was supported by the National Natural Science Foundation of China (Nos. 11731007, 11688101) and Innovation Grant from National Center for Mathematics and Interdisciplinary Sciences.

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Abidi, H., Zhang, P. On the Global Well-Posedness of 3-D Boussinesq System with Variable Viscosity. Chin. Ann. Math. Ser. B 40, 643–688 (2019). https://doi.org/10.1007/s11401-019-0156-2

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Keywords

  • Boussinesq systems
  • Littlewood-Paley theory
  • Variable viscosity
  • Maximal regularity of heat equation

2000 MR Subject Classification

  • 35Q30
  • 76D03