Skip to main content
SpringerLink
Log in

Some remarks on planar Boussinesq equations

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

Abstract

The main purpose of this paper is to prove the well-posedness of the two-dimensional Boussinesq equations when the initial vorticity ω 0 ∈ L 1(R 2) (or the finite Radon measure space). Using the stream function form of the equations and the Schauder fixed-point theorem to get the new proof of these results, we get that when the initial vorticity is smooth, there exists a unique classical solutions for the Cauchy problem of the two dimensional Boussinesq equations.

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. Beirão da Veiga, H. A new regularity class for the Navier-Stokes equations in Rn. Chin. Ann. of Math., 16B(4): 407–412 (1995)

    Google Scholar 

  2. Ben-Artzi, M. Global solutions of two-dimensional Navier-Stokes and Euler equations. Arch. Rat. Mech. Anal., 128: 329–358 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ben-Artzi, M. Planar Navier-Stokes equations vorticity approach. Handbook of mathematical fluid dynamics, North-Holland, Amsterdam

  4. Chae, D., Nam, H.-S. Local existence and blow-up criterion for the Boussinesq equations. Nagoya Math. J., 155: 55–80 (1999)

    MathSciNet  MATH  Google Scholar 

  5. Chae, D., Kim, S., Nam, H.-S. Local existence and blow-up criterion of Hölder continuous solutions of the Boussinesq equations. Nogaga Math. J., 155: 55–80 (1999)

    MathSciNet  MATH  Google Scholar 

  6. Kagei, Y. On weak solutions of nonstationary Boussinesq equations. Diff and Integral Equ, 9: 587–611 (1993)

    MathSciNet  Google Scholar 

  7. Kato, T. Strong L p solutions of the Navier-Stokes equations in R n, with application to weak solutions. Math. Z., 187: 471–480 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  8. Leray, J. Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math., 63: 193–248 (1934)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lions, P.L. Mathematical topics in fluid mechanics, Vol.1: incompressible models. In: Oxford Lecture Series in Mathematics and Its Applications, Vol.10, 1996

  10. McGrath, F.J. Nonstationary plane flow of viscous and ideal fluids. Arch. Rat. Mech. Anal., 27: 328–348 (1968)

    Article  MathSciNet  Google Scholar 

  11. Masuda, K. Weak solutions of the Navier-Stokes equations. Tohoku Math. J., 36: 623–646 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  12. Sohr, H. The Navier-Stokes Equations:an elementary functional analytic approach. Birkhäuser Verlag, 2001

  13. Temam, R. Navier-Stokes equations theory and numerical Analysis. North-Holland-Amsterdam, New York, 1984

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Beijing Technology and Business University, Beijing, 100048, China

    Xiao-jing Cai

  2. College of Applied Sciences, Beijing University of Technology, Beijing, 100124, China

    Xiao-jing Cai

  3. Department of Mathematics, Beijing Information Science and Technology University, Beijing, 100101, China

    Chun-yan Xue

  4. Institute of Mechanics, Chinese Academy of Sciences, Beijing, 100190, China

    Xian-jin Li

  5. Fundamental Department of the Academy of Armored Force Engineering, Beijing, 100072, China

    Ying Liu

  6. School of Mathematical Sciences, Capital Normal University, Beijing, 100048, China

    Quan-sen Jiu

Authors
  1. Xiao-jing Cai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chun-yan Xue
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xian-jin Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ying Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Quan-sen Jiu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiao-jing Cai.

Additional information

The first author is supported by the National Natural Science Foundation of China (No. 11171229); The third author is supported by 973 program (Grant No. 2011CB711100).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cai, Xj., Xue, Cy., Li, Xj. et al. Some remarks on planar Boussinesq equations. Acta Math. Appl. Sin. Engl. Ser. 28, 525–534 (2012). https://doi.org/10.1007/s10255-012-0167-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10255-012-0167-1

Keywords

2000 MR Subject Classification

Search

Navigation