Skip to main content
SpringerLink
Log in

Long time behavior for the equation of finite-depth fluids

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

In this paper we study the Cauchy problem for the generalized equation of finite-depth fluids

$$\partial _t u - G(\partial _x^2 u) - \partial _x \left( {\frac{{u^p }}{p}} \right) = 0$$

whereG(·) is a singular integral, andp is an integer larger than 1. We obtain the long time behavior of the fundamental solution of linear problem, and prove that the solutions of the nonlinear problem with small initial data for\(p > 5/2 + \sqrt {21} /2\) are decay in time and freely asymptotic to solutions of the linear problem. In addition we also study some properties of the singular integralG(·) inL q(R) withq>1.

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.

Similar content being viewed by others

References

  1. Abdelouhab, L., Bona, J.L., Felland, M., Saut, J.C.: Physica D40, 360–392 (1989)

    Google Scholar 

  2. Case, K.M.: Properties of the Benjamin-One equation. J. Math. Phys.20, 972–977 (1979)

    Article  Google Scholar 

  3. Ginibre, J., Tsutsumi, Y.: Uniqueness of solutions for the generalized KdV equation. SIAM J. Math. Anal20, (6), 1388–1425 (1989)

    Article  Google Scholar 

  4. Guo Boling, Tan Shaobin: On smooth solution to the initial value problem for the mixed nonlinear Schrodinger equations. Proc. Roy. Soc. Edinburgh119A, 31–45 (1991)

    Google Scholar 

  5. Joseph, R.I.: Solitary waves in a finite depth fluid. J. Phys.A, 10, L225-L227 (1977)

    Google Scholar 

  6. Kenig, C.E., Ponce, G., Vega, L.: On the generalized KdV equation. Duke Math. J,59, (3), 585–610 (1989)

    Article  Google Scholar 

  7. Klainerman, S., Ponce, G.: Global small amplitude solution to non-linear evolution equations. Commun. Pure Appl. Math.37, 133–141 (1983)

    Google Scholar 

  8. Iorio, R.: On the Cauchy problem for the Benjamin-Ono equation. Comm. PDE11, 1031–1081 (1986)

    Google Scholar 

  9. Ponce, G.: Smoothing properties of solutions to the Benjamin-Ono equation. Diff. and Integral Equations4, (3), 527–542 (1991)

    Google Scholar 

  10. Satsuma, J., Ablowitz, M.J., Kodama, Y.: On an internal Wave equation describing a stratified fluid with finite depth. Phys. Lett. A73, 283–286 (1979)

    Article  Google Scholar 

  11. Shatah, J.: Global existence of small solutions to nonlinear evolution equations. J. Diff. Eqs.46, 609–625 (1982)

    Article  Google Scholar 

  12. Sidi, A., Sulem, C., Sulem, P. L.: On the long time behavior of a generalized KdV equation. Acta Appl. Math.7, 35–47 (1986)

    Article  Google Scholar 

  13. Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton, NJ: Princeton University Press, 1970

    Google Scholar 

  14. Stein, E. M.: Oscillatory integrals in Fourier analysis. Beijing Lecture in Harmonic Analysis. Princeton, NJ: Princeton University Press 1986, pp. 307–355

    Google Scholar 

  15. Strauss, W. A.: Nonlinear scattering theory of low energy. J. Funct. Anal.41, 110–133 (1981)

    Article  Google Scholar 

  16. Zhou Yulin, Guo Boling,: Initial value problem for a nonlinear singular Integral-Differential equation of deep water. Lecture Notes in Math.1306, pp. 278–290

  17. Zhou Yulin, Guo Boling, Tan Shaobin: On the Cauchy problem for the equation of finite-depth fluids. J. PDE5, (1), 1–16 (1992)

    Google Scholar 

Download references

Author information

Author notes

  1. Tan Shaobin

    Present address: Department of Mathematics and Statistics, University of Saskatchewan, S7N OWO, Saskatoon, Canada

Authors and Affiliations

  1. Institute of Applied Physics and Computational Mathematics, P. O. Box 8009, 100088, Beijing, P. R. China

    Guo Boling & Tan Shaobin

Authors
  1. Guo Boling
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tan Shaobin
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by H. Araki

Rights and permissions

Reprints and permissions

About this article

Cite this article

Boling, G., Shaobin, T. Long time behavior for the equation of finite-depth fluids. Commun.Math. Phys. 163, 1–15 (1994). https://doi.org/10.1007/BF02101732

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02101732

Keywords

Search

Navigation