Advertisement

Communications in Mathematical Physics

, Volume 265, Issue 1, pp 189–200 | Cite as

Existence and Uniqueness of Low Regularity Solutions for the Dullin-Gottwald-Holm Equation

  • Octavian G. MustafaEmail author
Article

Abstract

We establish the local existence and uniqueness of solutions for the Dullin-Gottwald-Holm equation with continuously differentiable, periodic initial data. The regularity conditions needed for the Cauchy problem via the semigroup approach of quasilinear hyperbolic equations of evolution or the viscosity method are significantly lowered.

Keywords

Viscosity Neural Network Statistical Physic Complex System Initial Data 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)CrossRefADSzbMATHMathSciNetGoogle Scholar
  2. 2.
    Constantin, A., Escher, J.: Global existence and blow-up for a shallow water equation. Ann. Sc. Norm. Sup. Pisa 26, 303–328 (1998)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Constantin, A., McKean, H.P.: A shallow water equation on the circle. Comm. Pure Appl. Math. 52, 949–982 (1999)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Constantin, A., Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181, 229–243 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Constantin, A., Escher, J.: Well-posedness, global existence and blowup phenomena for a periodic quasi-linear hyperbolic equation. Comm. Pure Appl. Math. 61, 475–504 (1998)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Constantin, A., Molinet, L.: Global weak solutions for a shallow water equation. Commun. Math. Phys. 211, 45–61 (2000)CrossRefADSzbMATHMathSciNetGoogle Scholar
  7. 7.
    Constantin, A., Strauss, W.A.: Stability of peakons. Comm. Pure Appl. Math. 53, 603–610 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Constantin, A.: Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Ann. Inst. Fourier 50, 49–73 (2000)MathSciNetGoogle Scholar
  9. 9.
    Constantin, A., Strauss, W.A.: Stability of the Camassa-Holm solitons. J. Nonlinear Sci. 12, 415–422 (2002)CrossRefADSzbMATHMathSciNetGoogle Scholar
  10. 10.
    Constantin, A., Kolev, B.:On the geometric approach to the motion of inertial mechanical systems. J. Phys. A 35, R51–R79 (2002)Google Scholar
  11. 11.
    Constantin, A., Kolev, B.: Geodesic flow on the diffeomorphism group of the circle. Comment. Math. Helv. 78, 787–804 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Dai, H.H., Huo, Y.: Solitary shock waves and other travelling waves in a general compressible hyperelastic rod. Proc. Roy. Soc. London 456, 331–363 (2000)ADSzbMATHCrossRefGoogle Scholar
  13. 13.
    Degasperis, A., Procesi, M.: Asymptotic integrability. Symmetry and perturbation theory, River Edge, NJ: World Sci. Publishing, 1999, pp. 23–37Google Scholar
  14. 14.
    Dullin, H.R., Gottwald, G., Holm, D.D.: An integrable shallow water equation with linear and nonlinear dispersion. Phys. Rev. Lett. 87, 4501–4504 (2001)CrossRefADSGoogle Scholar
  15. 15.
    Dullin, H.R., Gottwald, G., Holm, D.D.: Camassa-Holm, Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow water waves. Fluid Dynam. Res. 33, 73–95 (2003)CrossRefADSzbMATHMathSciNetGoogle Scholar
  16. 16.
    Guo, B.L., Liu, Z.R.: Peaked wave solutions of CH – γ equation. Sci. China, Ser. A 33, 325–337 (2003)Google Scholar
  17. 17.
    Johnson, R.S.: A modern introduction to the mathematical theory of water waves. Cambridge: Cambridge Univ. Press, 1997Google Scholar
  18. 18.
    Johnson, R.S.: Camassa-Holm, Korteweg-de Vries and related models for water waves. J. Fluid Mech. 457, 63–82 (2002)CrossRefADSGoogle Scholar
  19. 19.
    Kato, T.: Quasi-linear equations of evolution, with aplications to partial differential equations. In: Spectral theory and differential equations, Lecture Notes in Mathematics, Vol. 448, Berlin: Springer-Verlag, 1975, pp. 25–70Google Scholar
  20. 20.
    Kato, T.: On the Korteweg-de Vries equation. Manuscripta Math. 28, 89–99 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Kato, T.: Abstract evolution equations, linear and quasilinear, revisited. In: Functional analysis and related topics, Lecture Notes in Mathematics, Vol. 1540, New York: Springer-Verlag, 1993, pp. 103–125Google Scholar
  22. 22.
    Lenells, J.: Stability of periodic peakons. Int. Math. Res. Not. 10, 485–499 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Lenells, J.: A variational approach to the stability of periodic peakons. J. Nonlinear Math. Phys. 11, 151–163 (2004)zbMATHMathSciNetADSGoogle Scholar
  24. 24.
    Mustafa, O.G.: On the Cauchy problem for a generalized Camassa-Holm equation. Nonlinear Anal. 64, 1382–1399 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    Li, Y., Olver, P.: Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Differ. Eqs. 162, 27–63 (2000)CrossRefzbMATHGoogle Scholar
  26. 26.
    Pazy, A.: Semigroups of linear operators and applications to partial differential equations. New York: Springer-Verlag, 1983Google Scholar
  27. 27.
    Rodríguez-Blanco, G.: On the Cauchy problem for the Camassa-Holm equation. Nonlinear Anal. 46, 309–327 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Tang, M., Yang, C.: Extension on peaked wave solutions of CH – γ equation. Chaos Sol. Fract. 20, 815–825 (2004)CrossRefzbMATHGoogle Scholar
  29. 29.
    Tian, L., Gui, G., Liu, Y.: On the well-posedness problem and the scattering problem for the Dullin-Gottwald-Holm equation. Commun. Math. Phys. 257, 667–701 (2005)CrossRefADSzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CraiovaCraiovaRomania

Personalised recommendations