Global well-posedness of a model on 2D Boussinesq–Bénard equations

Abstract

In this paper, we consider the classical solutions to a model of two-dimensional incompressible inviscid Boussinesq–Bénard equations. Notice that, in the case when the source term of temperature equation in this model is the second component of velocity \(u_2\) or no source term, there is no global-in-time existence result for the general initial data. Here, if the source term is only chosen as \(\Delta u_2\), then we can obtain the global well-posedness, inviscid limit and some exponential decay estimates. Our key observation is the nice symmetrical structure hidden in the corresponding system, which plays an extremely important role in the global well-posedness studied here.

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Acknowledgements

Xu was partially supported by the National Natural Science Foundation of China (Nos. 11771045, 11871087) and the Natural Science Foundation of Beijing (No. 2112023) and by the Fundamental Research Funds for the Central Universities of China. Ye was supported by the National Natural Science Foundation of China (No. 11701232) and the Qing Lan Project of Jiangsu Province.

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Correspondence to Zhuan Ye.

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Li, C., Xu, X. & Ye, Z. Global well-posedness of a model on 2D Boussinesq–Bénard equations. Z. Angew. Math. Phys. 72, 18 (2021). https://doi.org/10.1007/s00033-020-01456-9

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Keywords

  • Boussinesq–Bénard equations
  • Global regularity
  • Decay estimates

Mathematics Subject Classification

  • 35Q35
  • 35B65
  • 76D03