Advertisement

Acta Mathematica Sinica

, Volume 22, Issue 5, pp 1457–1466 | Cite as

Global Existence of Solutions for the Cauchy Problem of the Kawahara Equation with L 2 Initial Data

  • Shang Bin Cui
  • Dong Gao Deng
  • Shuang Ping Tao
ORIGINAL ARTICLES

Abstract

In this paper we study solvability of the Cauchy problem of the Kawahara equation \( \partial _{t} u + au\partial _{x} u + \beta \partial ^{3}_{x} u + \gamma \partial ^{5}_{x} u = 0 \) with L 2 initial data. By working on the Bourgain space X r,s (R 2) associated with this equation, we prove that the Cauchy problem of the Kawahara equation is locally solvable if initial data belong to H r (R) and −1 < r ≤ 0. This result combined with the energy conservation law of the Kawahara equation yields that global solutions exist if initial data belong to L 2(R).

Keywords

Kawahara equation Cauchy problem global solution 

MR (2000) Subject Classification

35Q53 35Q35 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Kawahara, T.: Oscillatory solitary waves in dispersive media. J. Phys. Soc. Japan, 33, 260–264 (1972)CrossRefGoogle Scholar
  2. 2.
    Gorshkov, K. A., Papko, V. V.: The structure of solitary waves in media with anomalously small dispersion. Sov. Phys. JETP, 46, 92–96 (1977)Google Scholar
  3. 3.
    Abramyan, L. A., Stepanyants, Yu. A.: The structure of two–dimensional solitons in media with anomalously small dispersion. Sov. Phys. JETP, 61, 963–966 (1985)Google Scholar
  4. 4.
    Karpman, V. I., Belashov, V. Yu.: Dynamics of two–dimensional soliton in weakly dispersive media. Phys. Lett. A., 154, 131–139 (1991)CrossRefGoogle Scholar
  5. 5.
    Hunter, J. K., Scheurle, J.: Existence of perturbed solitary wave solutions to a model equation for water waves. Physica D, 32, 253–268 (1988)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Pomeau, Y., Ramani, A., Grammaticos, B.: Structural stability of the Korteweg–de Vries solitons under a singular perturbation. Physica D, 31, 127–134 (1988)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Boyd, J. P.: Weakly non–local solitons for capillary–gravity waves: fifth degree Korteweg–de Vries equation. Phys. D, 48, 129–146 (1991)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Il’ichev, A. T., Semenov, A. Yu.: Stability of solitary waves in dispersive media described by a fifth order evolution equation. Theor. Comput. Fluid Dynamics, 3, 307–326 (1992)CrossRefGoogle Scholar
  9. 9.
    Kichenassamy, S., Olver, P. J.: Existence and nonexistence of solitary wave solutions to higher–order model evolution equations. SIAM J. Math. Anal., 23, 1141–1166 (1992)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Grimshaw, R., Joshi, N.: Weakly nonlocal solitary waves in a singularly perturbed Kortweg–de Vries equation. SIAM J. Appl. Math., 55, 124–135 (1995)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Cui, S., Tao, S.: Strichartz estimates for dispersive equations and solvability of Cauchy problems of the Kawahara equation. J. Math. Anal. Appl., to appearGoogle Scholar
  12. 12.
    Bourgain, J.: Fourier transformation restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations I: Schrödinger equations. Geom. Func. Anal., 3, 107–156 (1993)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Bourgain, J.: Fourier transformation restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations II: The KdV–equation. Geom. Func. Anal., 3, 209–262 (1993)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Bourgain, J.: On the Cauchy problem for the Kadomstev–Petviashvili equation. Geom. Func. Anal., 3, 315–341 (1993)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Kenig, C. E., Ponce, G., Vega, L.: The Cauchy problem for the Korteweg–de Vries equation in Sobolev spaces of negative indices. Duke Math. J., 71, 1–21 (1993)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Kenig, C. E., Ponce, G., Vega, L.: A bilinear estimate with applications to the KdV equation. J. Amer. Math. Soc., 9, 573–603 (1996)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Kenig, C. E., Ponce, G., Vega, L.: Well–posedness of the initial value problem for the Korteweg–de Vries equation. J. Amer. Math. Soc., 4, 323–347 (1991)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Kenig, C. E., Ponce, G., Vega, L.: Well–posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle. Comm. Pure Appl. Math., 46, 527–620 (1993)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Shang Bin Cui
    • 1
  • Dong Gao Deng
    • 1
  • Shuang Ping Tao
    • 2
  1. 1.Department of MathematicsSun Yat-Sen UniversityGuangzhou 510275P. R. China
  2. 2.Department of MathematicsNorthwest Normal UniversityLanzhou 730070P. R. China

Personalised recommendations