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Mathematical Physics, Analysis and Geometry

, Volume 15, Issue 4, pp 317–329 | Cite as

On a Shallow Water Equation Perturbed in Schwartz Class

  • Xiang’ou ZhuEmail author
Article

Abstract

We discuss the Camassa-Holm equation perturbed in Schwartz class around a suitable constant. This paper is concerned with the wave breaking mechanism for periodic case where two special classes of initial data were involved. The asymptotic behavior of solutions is also analyzed in the following sense: the corresponding solution to initial data with algebraic decay at infinity will retain this property at infinity in its lifespan.

Keywords

The Camassa-Holm equation Singularity Asymptotic behavior 

Mathematics Subject Classifications (2010)

37L05 35Q58 35L30 35Q35 

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References

  1. 1.
    Boutet de Monvel, A., Kostenko, A., Shepelsky, D., Teschl, G.: Long-time asymptotics for the Camassa-Holm equation. SIAM J. Math. Anal. 41, 1559–1588 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bressan, A., Constantin, A.: Global conservative solutions of the Camassa-Holm equation. Arch. Ration. Mech. Anal. 183, 215–239 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bressan, A., Constantin, A.: Global dissipative solutions of the Camassa-Holm equation. Anal. Appl. 5, 1–27 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Camassa, R., Holm, D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)MathSciNetADSzbMATHCrossRefGoogle Scholar
  5. 5.
    Constantin, A.: On the scattering problem for the Camassa-Holm equation. Proc. R. Soc. Lond. A 457, 953–970 (2001)MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. 6.
    Constantin, A.: Finite propagation speed for the Camassa-Holm equation. J. Math. Phys. 46, 023506 (2005)MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    Constantin, A.: The trajectories of particles in Stokes waves. Invent. Math 166, 523–535 (2006)MathSciNetADSzbMATHCrossRefGoogle Scholar
  8. 8.
    Constantin, A., Lannes, D.: The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations. Arch. Ration. Mech. Anal. 192, 165–186 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Constantin, A., Escher, J.: Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation. Commun. Pure Appl. Math. 51, 475–504 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Constantin, A., Escher, J.: Particle trajectories in solitary water waves. Bull. Am. Math. Soc. 44, 423–431 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Constantin, A., Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181, 229–243 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Constantin, A., Mckean, H.P.: A shallow water equation on the circle. Commun. Pure Appl. Math. 52, 949–982 (1999)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Constantin, A., Strauss, W.: Stability of peakons. Commun. Pure Appl. Math. 53, 603–610 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Constantin, A., Gerdjikov, V.S., Ivanov, R.I.: Inverse scattering transform for the Camassa-Holm equation. Inverse Probl. 22, 2197–2207 (2006)MathSciNetADSzbMATHCrossRefGoogle Scholar
  15. 15.
    Fuchssteiner, B., Fokas, A.: Symplectic structures, their Backlund transformations and hereditary symmetries. Phys. D 4, 47–66 (1981/1982)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Guo, Z.: Blow up, global existence and infinite propagation speed for the weakly dissipative Camassa-Holm equation. J. Math. Phys. 49, 033516 (2008)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Guo, Z.: Some properties of solutions to the weakly dissipative Degasperis-Procesi equation. J. Differ. Equ. 246, 4332–4344 (2009)zbMATHCrossRefGoogle Scholar
  18. 18.
    Guo, Z.: Asymptotic profiles of solutions to the two-component Camassa-Holm system. Nonlinear Anal. 75, 1–6 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Guo, Z., Ni, L.: Wave breaking for the periodic weakly dissipative DGH equation. Nonlinear Anal. 74, 965–973 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Guo, Z., Zhou, Y.: Wave breaking and persistence properties for the dispersive Rod equation. SIAM J. Math. Anal. 40, 2567–2580 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Henry, D.: Compactly supported solutions of the Camassa-Holm equation. J. Nonlin. Math. Phys. 12, 342–347 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Henry, D.: Persistence properties for a family of nonlinear partial differential equations. Nonlinear Anal. 70, 1565–1573 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Himonas, A., Misiołek, G., Ponce, G., Zhou, Y.: Persistence properties and unique continuation of solutions of the Camassa-Holm equation. Commun. Math. Phys. 271, 511–522 (2007)ADSzbMATHCrossRefGoogle Scholar
  24. 24.
    Ivanov, R.I.: Water waves and integrability. Philos. Trans. R. Soc. Lond. A 365, 2267–2280 (2007)ADSzbMATHCrossRefGoogle Scholar
  25. 25.
    Jiang, Z., Ni, L., Zhou, Y.: Wave breaking for the Camassa-Holm equation. J. Nonlinear Sci. 22, 235–245 (2012). doi: 10.1007/s00332-011-9115-0 MathSciNetADSzbMATHCrossRefGoogle Scholar
  26. 26.
    Johnson, R.S.: Camassa-Holm, Korteweg-de Vries and related models for water waves. J. Fluid Mech. 455, 63–82 (2002)MathSciNetADSzbMATHCrossRefGoogle Scholar
  27. 27.
    Kato, T.: Quasi-linear equations of evolution with application to partial differential equations. In: Spectral Theory and Differential Equations (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jörgens). Lecture Notes in Math., vol. 448, pp. 25–70. Springer, Berlin (1975)Google Scholar
  28. 28.
    Li, Y., Olver, P.: Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Differ. Equ. 162, 27–63 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    McKean, H.P.: Breakdown of a shallow water equation. Asian J. Math. 2, 767–774 (1998)MathSciNetGoogle Scholar
  30. 30.
    Misiolek, G.: Classical solutions of the periodic Camassa-Holm equation. Geom. Funct. Anal. 12, 1080–1104 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Molinet, L.: On well-posedness results for Camassa-Holm equation on the line: a survey. J. Nonlin. Math. Phys. 11, 521–533 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Ni, L., Zhou, Y.: Wave breaking and propagation speed for a class of nonlocal dispersive θ-equations. Nonlinear Anal.: Real World Appl. 12, 592–600 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Tian, L., Gui, G., Liu, Y.: On the Cauchy problem and the scattering problem for the Dullin-Gottwald-Holm equation. Commun. Math. Phys. 257, 667–701 (2005)MathSciNetADSzbMATHCrossRefGoogle Scholar
  34. 34.
    Toland, J.F.: Stokes waves. Topol. Methods Nonlinear Anal. 7, 1–48 (1996)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Whitham, G.B.: Linear and Nonlinear Waves. Wiley-Interscience, New York-London-Sydney (1974)zbMATHGoogle Scholar
  36. 36.
    Xin, Z., Zhang, P.: On the weak solutions to a shallow water equation. Commun. Pure Appl. Math. 53, 1411–1433 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Zhou, Y.: Wave breaking for a shallow water equation. Nonlinear Anal. 57, 137–152 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Zhou, Y.: Wave breaking for a periodic shallow water equation. J. Math. Anal. Appl. 290, 591–604 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Zhou, Y.: Blow-up of solutions to a nonlinear dispersive rod equation. Calc. Var. Partial Differ. Equ. 25, 63–77 (2006)zbMATHCrossRefGoogle Scholar
  40. 40.
    Zhou, Y.: On solutions to the Holm-Staley b-family of equations. Nonlinearity 23, 369–381 (2010)MathSciNetADSzbMATHCrossRefGoogle Scholar
  41. 41.
    Zhou, Y., Chen, H.: Wave breaking and propagation speed for the Camassa–Holm equation with k ≠ 0. Nonlinear Anal.: Real World Appl. 12, 1875–1882 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Zhou, Y., Guo, Z.: Blow up and propagation speed of solutions to the DGH equation. Discrete Contin. Dyn. Syst. Ser. B 12, 657–670 (2009)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.College of Physics and Electronic Information EngineeringWenzhou UniversityWenzhouChina
  2. 2.The Key Laboratory of Low-voltage Apparatus IntellectualTechnology of ZhejiangWenzhouChina

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