Skip to main content
SpringerLink
Log in

Analytical Smoothing Effect of Solution for the Boussinesq Equations

  • Published:
Acta Mathematica Scientia Aims and scope Submit manuscript

Abstract

In this article, we study the analytical smoothing effect of Cauchy problem for the incompressible Boussinesq equations. Precisely, we use the Fourier method to prove that the Sobolev H1-solution to the incompressible Boussinesq equations in periodic domain is analytic for any positive time. So the incompressible Boussinesq equations admit exactly same smoothing effect properties of incompressible Navier-Stokes 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. Abidi H, Hmidi T. On the global well posedness for Boussinesq system. J Diff Equ, 2007, 233(1): 199–220

    Article  MathSciNet  MATH  Google Scholar 

  2. Biswas A. Gevrey regularity for a class of dissipative equations with applications to decay. J Differential Equations, 2012, 253: 2739–2764

    Article  MathSciNet  MATH  Google Scholar 

  3. Brandolese L, Schonbek E. Large time decay and growth for solutions of a viscous Boussinesq system. Trans Amer Math Soc, 2012, 364: 5057–5090

    Article  MathSciNet  MATH  Google Scholar 

  4. Guo Boling, Wang Bixiang. Gevrey class regularity and approximate inertial manifolds for the Newton-Boussinesq equations. Chin Ann Math, Series B, 1998, 19(2): 179–188

    MathSciNet  MATH  Google Scholar 

  5. Cannon J R, DiBenedetto E. The initial problem for the Boussinesq equations with data in L p// Lecture Notes in Mathematics, Vol 771. Berlin: Springer, 1980: 129–144

    Article  MATH  Google Scholar 

  6. Cheng Feng, Li Wei-Xi, Xu Chao-Jiang. Gevrey regularity with weight for incompressible Euler equation in the half plane. Acta Math Sci, 2017, 37B(4): 1115–1132

    Article  MathSciNet  MATH  Google Scholar 

  7. Chae D, Nam H-S. Local existence and blow-up criterion for the Boussinesq equations. Proc Roy Soc Edinburgh, Sect A, 1997, 127(5): 935–946

    Article  MathSciNet  MATH  Google Scholar 

  8. Chae D, Kin S-K, Nam H-S. Local existence and blow-up criterion of Hölder continuous solutions of the Boussinesq equations. Nagoya Math J, 1999, 155: 55–80

    Article  MathSciNet  MATH  Google Scholar 

  9. Chae D. Global regularity for the 2D Boussinesq equations with partial viscosity terms. Adv Math, 2006, 203: 497–513

    Article  MathSciNet  MATH  Google Scholar 

  10. Danchin, R; Paicu, M. Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces. Phys D, 2008, 237(10/12): 1444–1460

    Article  MathSciNet  MATH  Google Scholar 

  11. Danchin R, Paicu M. Global well-posedness issue for the inviscid Boussinesq system with Youdovich’s type data. Comm Math Phys, 2009, 290(1): 1–14

    Article  MathSciNet  MATH  Google Scholar 

  12. Díaz J I, Galiano G. Existence and uniqueness of solutions of the Boussinesq system with nonlinear thermal diffusion. Topol Methods Nonlinear Anal, 1998, 11(1): 59–82

    Article  MathSciNet  MATH  Google Scholar 

  13. Foias C, Temam R. Gevrey class regularity for the solutions of the Navier-Stokes equations. J Funct Anal, 1989, 87: 359–369

    Article  MathSciNet  MATH  Google Scholar 

  14. Gill A E. Atmosphere-Ocean Dynamics. London: Academic Press, 1982

    Google Scholar 

  15. Hmidi T, Keraani S. On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity. Adv Differential Equations, 2007, 12(4): 461–480

    MathSciNet  MATH  Google Scholar 

  16. Hmidi T, Keraani S. On the global well-posedness of the Boussinesq system with zero viscosity. Indiana Univ Math J, 2011, 58(4): 1591–1618

    Article  MathSciNet  MATH  Google Scholar 

  17. Levermore C D, Oliver M. Analyticity of solutions for a generalized Euler equation. J Diff Equ, 1997, 133: 321–339

    Article  MathSciNet  MATH  Google Scholar 

  18. Majda A J, Bertozzi A L. Vorticity and Incompressible Flow. Cambridge University Press, 2002

    MATH  Google Scholar 

  19. Majda A J. Introduction to PDEs and Waves for the Atmosphere and Ocean. Courant Lecture Notes in Mathematics 9, AMS/CIMS, 2003

    Book  MATH  Google Scholar 

  20. Pedlosky J. Geophysical Fluid Dyanmics. New York: Springer-Verlag, 1987

    Book  MATH  Google Scholar 

  21. Paicu M, Vicol V. Analyticity and Gevrey-class regularity for the second-grade fluid equations. J Math Fluid Mech, 2011, 13: 533–555

    Article  MathSciNet  MATH  Google Scholar 

  22. Shu C E W. Small scale structure un Boussinesq convection. Phys Fluids, 1994, 6: 48–54

    Google Scholar 

  23. Hou T Y, Li C. Global well-posedness of the viscous Boussinesq equations. Discrete and Continuous Dynalical Systems, 2005, 12(1): 1–12

    MathSciNet  MATH  Google Scholar 

  24. Sawada O, Taniuchi Y. On the boussinesq flow with nondecaying initial data. Funk Ekvac, 2004, 47(2): 225–250

    Article  MathSciNet  MATH  Google Scholar 

  25. Temam R. Navier-Stokes Equations and Nonlinear Functional Analysis, Vol 66. SIAM, 1995

Download references

Author information

Authors and Affiliations

  1. Hubei Key Laboratory of Applied Mathematics; School of Mathematics and Statistics, Hubei University, Wuhan, 430062, China

    Feng Cheng  (程峰)

  2. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Chaojiang Xu  (徐超江)

  3. Université de Rouen, CNRS UMR 6085, Laboratoire de Mathématiques, 76801, Saint-Etienne du Rouvray, France

    Chaojiang Xu  (徐超江)

Authors
  1. Feng Cheng  (程峰)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chaojiang Xu  (徐超江)
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Feng Cheng  (程峰).

Additional information

The research of the second author was supported partially by “The Fundamental Research Funds for Central Universities of China”.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cheng, F., Xu, C. Analytical Smoothing Effect of Solution for the Boussinesq Equations. Acta Math Sci 39, 165–179 (2019). https://doi.org/10.1007/s10473-019-0114-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10473-019-0114-9

Key words

2010 MR Subject Classification

Search

Navigation