Optimal Decay Estimates for 2D Boussinesq Equations with Partial Dissipation

Abstract

Buoyancy-driven fluids such as many atmospheric and oceanic flows and the Rayleigh–Bénard convection are modeled by the Boussinesq systems. By rigorously estimating the large-time behavior of solutions to a special Boussinesq system, this paper reveals a fascinating phenomenon on buoyancy-driven fluids that the temperature can actually stabilize the fluids. The Boussinesq system concerned here governs the motion of perturbations near the hydrostatic equilibrium. When the buoyancy forcing is not present, the velocity of the fluid obeys the 2D Navier–Stokes equation with only vertical dissipation and its Sobolev norm could potentially grow even though its precise large-time behavior remains open. This paper shows that the temperature through the coupling and interaction tames and regularizes the fluids, and causes the velocity (measured in Sobolev norms) to decay in time. Optimal decay rates are obtained.

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Acknowledgements

S. Lai was partially supported by the National Natural Science Foundation of China (Nos. 11871407, 12071390). J. Wu was partially supported by the National Science Foundation of USA under grant DMS 1624146 and the AT&T Foundation at Oklahoma State University. X. Xu was partially supported by the National Natural Science Foundation of China (No. 11771045). J. Zhang was partly supported by the National Natural Science Foundation of China (Nos. 11671333, 12071390). Y. Zhong was partially supported by the National Natural Science Foundation of China (Nos. 11771043, 11771045).

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Lai, S., Wu, J., Xu, X. et al. Optimal Decay Estimates for 2D Boussinesq Equations with Partial Dissipation. J Nonlinear Sci 31, 16 (2021). https://doi.org/10.1007/s00332-020-09672-3

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Keywords

  • Boussinesq equations
  • Optimal decay estimate
  • Partial dissipation

Mathematics Subject Classification

  • 35Q35
  • 35B40
  • 42A38
  • 76D50