Skip to main content
SpringerLink
Log in

Global Well-posedness for the 2D Micropolar Rayleigh-Bénard Convection Problem without Velocity Dissipation

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

In this article, we study the Cauchy problem to the micropolar Rayleigh-Bénard convection problem without velocity dissipation in two dimension. We first prove the local well-posedness of a smooth solution, and then establish a blow up criterion in terms of the gradient of scalar temperature field. At last, we obtain the global well-posedness to the system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Amann, H.: Maximal regularity for nonautonoumous evolution equations. Adv. Nonlinear Stud., 4, 417–430 (2004)

    Article  MathSciNet  Google Scholar 

  2. Aubin, J.: Un théorème de compacité. Comptes Rendus de l’Académie des Sciences, Paris, 256, 5042–5044 (1963)

    MathSciNet  MATH  Google Scholar 

  3. Bahouri, H., Chemin, J. Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Heidelberg, 2011

    Book  Google Scholar 

  4. Beale, J., Kato, T., Majda, A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Comm. Math. Phys., 94, 61–66 (1984)

    Article  MathSciNet  Google Scholar 

  5. Chemin, J. Y.: Perfect incompressible fluids, Oxford University Press, New York, 1998

    MATH  Google Scholar 

  6. Dong, B., Zhang, Z.: Global regularity of the 2D micropolar fluid flows with zero angular viscosity. J. Differential Equations, 249, 200–213 (2010)

    Article  MathSciNet  Google Scholar 

  7. Dong, B., Li, J., Wu, J.: Global well-posedness and large-time decay for the 2D micropolar equations. J. Differential Equations, 262, 3488–3523 (2017)

    Article  MathSciNet  Google Scholar 

  8. Eringen, A. C.: Theory of micropolar fluids. J. Math. Mech., 16, 1–18 (1996)

    MathSciNet  Google Scholar 

  9. Foias, C., Manley, O., Temam, R.: Attractors for the Béenard problem: existence and physical bounds on their fractal dimension. Nonlinear Analysis (TMA), 11(8), 939–967 (1987)

    Article  Google Scholar 

  10. Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier-Stokes equations. Comm. Pure Appl. Math., 41(7), 891–907 (1988)

    Article  MathSciNet  Google Scholar 

  11. Kalita, P., Langa, J., Łukaszewicz, G.: Micropolar meets Newtonian. Rayleigh-Bénard problem, Physica D., 392, 57–80 (2019)

    Article  Google Scholar 

  12. Popel, A., Regirer, S., Usick, P.: A continuum model of blood flow. Biorheology, 11, 427–437 (1974)

    Article  Google Scholar 

  13. Tarasinska, A.: Global attractor for heat convection problem in a micropolar fluid. Math. Meth. Appl. Sci., 29, 1215–1236 (2006)

    Article  MathSciNet  Google Scholar 

  14. Xue, L.: Well posedness and zero microrotation viscosity limit of the 2D micropolar fluid equations. Math. Methods Appl. Sci., 34, 1760–1777 (2011)

    Article  MathSciNet  Google Scholar 

  15. Xu, F., Chi, M.: Global regularity for the 2D micropolar Rayleigh-Bénard convection system with the zero diffusivity. Applied Mathematics Letters, 108, 106508 (2020)

    Article  MathSciNet  Google Scholar 

  16. Xu, F., Qiao, L., Zhang, M.: On the well-posedness for the 2D micropolar Rayleigh-Bénard convection problem. Z. Angew. Math. Phys., 72, 17–29 (2021)

    Article  Google Scholar 

Download references

Acknowledgements

We are grateful to Professor Yi Zhou for his guidance and encouragement, which greatly improved our original manuscript.

Author information

Authors and Affiliations

  1. Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, 200433, P. R. China

    Sheng Wang

Authors
  1. Sheng Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sheng Wang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, S. Global Well-posedness for the 2D Micropolar Rayleigh-Bénard Convection Problem without Velocity Dissipation. Acta. Math. Sin.-English Ser. 37, 1053–1065 (2021). https://doi.org/10.1007/s10114-021-1040-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-021-1040-z

Keywords

MR(2010) Subject Classification

Search

Navigation