Skip to main content
SpringerLink
Log in

Long time solutions for the 2D inviscid Boussinesq equations with strong stratification

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

We consider the initial value problem of the 2D inviscid Boussinesq equations for stably stratified fluids. We prove the long time existence of classical solutions for large initial data in \(H^s(\mathbb {R}^2)\) with \(s>2\) when the buoyancy frequency is sufficiently high. Furthermore, we consider the singular limit of the strong stratification, and show that the asymptotic profile of the long time solution is given by the corresponding linear dispersive solution.

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. Bona, J.L., Smith, R.: The initial-value problem for the Korteweg-de Vries equation. Philos. Trans. Roy. Soc. 278, 555–601 (1975)

    MathSciNet  Google Scholar 

  2. Cannon, J.R., DiBenedetto, E.: The Initial Value Problem for the Boussinesq Equations with Data in \(l^{p}\). Lecture Notes in Mathematics, pp. 129–144. Springer, Berlin (1980)

    Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Chae, D.: Local existence and blow-up criterion for the Euler equations in the Besov spaces. Asymptot. Anal. 38, 339–358 (2004)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Chen, Q., Miao, C., Zhang, Z.: On the well-posedness of the ideal MHD equations in the Triebel–Lizorkin spaces. Arch. Ration. Mech. Anal. 195, 561–578 (2010)

    Article  MathSciNet  Google Scholar 

  8. Christ, F.M., Weinstein, M.I.: Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation. J. Funct. Anal. 100, 87–109 (1991)

    Article  MathSciNet  Google Scholar 

  9. Danchin, R.: Remarks on the lifespan of the solutions to some models of incompressible fluid mechanics. Proc. Amer. Math. Soc. 141, 1979–1993 (2013)

    Article  MathSciNet  Google Scholar 

  10. Elgindi, T.M., Widmayer, K.: Sharp decay estimates for an anisotropic linear semigroup and applications to the surface quasi-geostrophic and inviscid Boussinesq systems. SIAM J. Math. Anal. 47, 4672–4684 (2015)

    Article  MathSciNet  Google Scholar 

  11. Ginibre, J., Velo, G.: Generalized Strichartz inequalities for the wave equation. J. Funct. Anal. 133, 50–68 (1995)

    Article  MathSciNet  Google Scholar 

  12. Greenleaf, A.: Principal curvature and harmonic analysis. Indiana Univ. Math. J. 30, 519–537 (1981)

    Article  MathSciNet  Google Scholar 

  13. Hou, T.Y., Li, C.: Global well-posedness of the viscous Boussinesq equations. Discret. Contin. Dyn. Syst. 12, 1–12 (2005)

    Article  MathSciNet  Google Scholar 

  14. Kato, T.: Nonstationary flows of viscous and ideal fluids in \({ R}^{3}\). J. Funct. Anal. 9, 296–305 (1972)

    Article  Google Scholar 

  15. Kato, T., Lai, C.Y.: Nonlinear evolution equations and the Euler flow. J. Funct. Anal. 56, 15–28 (1984)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. Keel, M., Tao, T.: Endpoint Strichartz estimates. Amer. J. Math. 120, 955–980 (1998)

    Article  MathSciNet  Google Scholar 

  18. Koh, Y., Lee, S., Takada, R.: Strichartz estimates for the Euler equations in the rotational framework. J. Differ. Equ. 256, 707–744 (2014)

    Article  MathSciNet  Google Scholar 

  19. Lee, S., Takada, R.: Dispersive estimates for the stably stratified Boussinesq equations. Indiana Univ. Math. J. 66, 2037–2070 (2017)

    Article  MathSciNet  Google Scholar 

  20. Liu, X., Wang, M., Zhang, Z.: Local well-posedness and blowup criterion of the Boussinesq equations in critical Besov spaces. J. Math. Fluid Mech. 12, 280–292 (2010)

    Article  MathSciNet  Google Scholar 

  21. Littman, W.: Fourier transforms of surface-carried measures and differentiability of surface averages. Bull. Am. Math. Soc. 69, 766–770 (1963)

    Article  MathSciNet  Google Scholar 

  22. Stein, E.M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton Mathematical Series. Princeton University Press, Princeton (1993)

    Google Scholar 

  23. Strichartz, R.: Restriction of Fourier transform to quadratic surfaces and decay of solutions to the wave equation. Duke Math J. 44, 705–714 (1977)

    Article  MathSciNet  Google Scholar 

  24. Takada, R.: Local existence and blow-up criterion for the Euler equations in Besov spaces of weak type. J. Evol. Equ. 8, 693–725 (2008)

    Article  MathSciNet  Google Scholar 

  25. Takada, R.: Long time existence of classical solutions for the 3D incompressible rotating Euler equations. J. Math. Soc. 68, 579–608 (2016)

    Article  MathSciNet  Google Scholar 

  26. Takada, R.: Strongly stratified limit for the 3D inviscid Boussinesq equations. Arch. Rational. Mech. Anal. 232(3), 1475–1503 (2019)

    Article  MathSciNet  Google Scholar 

  27. Taniuchi, Y.: Remarks on global solvability of 2-D Boussinesq equations with non-decaying initial data. Funkcial. Ekvac. 49, 39–57 (2006)

    Article  MathSciNet  Google Scholar 

  28. Tomas, P.A.: A restriction theorem for the Fourier transform. Bull. Am. Math. Soc. 81, 477–478 (1975)

    Article  MathSciNet  Google Scholar 

  29. Wan, R., Chen, J.: Global well-posedness for the 2D dispersive SQG equation and inviscid Boussinesq equations. Z. Angew. Math. Phys. 67(104), 1–22 (2016)

    MathSciNet  Google Scholar 

  30. Widmayer, K.: Convergence to stratified flow for an inviscid 3D Boussinesq system. Commun. Math. Sci. 16, 1713–1728 (2018)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by JSPS KAKENHI Grant Number JP15H05436.

Author information

Authors and Affiliations

  1. Faculty of Mathematics, Kyushu University, Fukuoka, 819-0395, Japan

    Ryo Takada

Authors
  1. Ryo Takada
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ryo Takada.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Takada, R. Long time solutions for the 2D inviscid Boussinesq equations with strong stratification. manuscripta math. 164, 223–250 (2021). https://doi.org/10.1007/s00229-019-01174-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-019-01174-1

Mathematics Subject Classification

Search

Navigation