Advertisement

Communications in Mathematical Physics

, Volume 267, Issue 3, pp 801–820 | Cite as

Global Existence and Blow-Up Phenomena for the Degasperis-Procesi Equation

  • Yue Liu
  • Zhaoyang Yin
Article

Abstract

This paper is concerned with several aspects of the existence of global solutions and the formation of singularities for the Degasperis-Procesi equation on the line. Global strong solutions to the equation are determined for a class of initial profiles. On the other hand, it is shown that the first blow-up can occur only in the form of wave-breaking. A new wave-breaking mechanism for solutions is described in detail and two results of blow-up solutions with certain initial profiles are established.

Keywords

Soliton Global Existence Travel Wave Solution Wave Breaking Shallow Water Equation 
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.
    Beals R., Sattinger D., Szmigielski J. (1998) Acoustic scattering and the extended Korteweg-de Vries hierarchy. Adv. Math. 140, 190–206MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bressan A., Constantin A.: Global conservative solutions of the Camassa-Holm equation. Preprint, www.math.ntnu.no/conservation/2006/023.html, To appear in Arch. Rat. Mech. Anal.Google Scholar
  3. 3.
    Camassa R., Holm D. (1993) An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664MATHCrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Camassa R., Holm D., Hyman J. (1994) A new integrable shallow water equation. Adv. Appl. Mech. 31, 1–33CrossRefGoogle Scholar
  5. 5.
    Coclite G.M., Karlsen K.H. (2006) On the well-posedness of the Degasperis-Procesi equation. J. Funct. Anal. 233, 60–91MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Coclite G.M., Karlsen K.H., Risebro N.H.: Numerical schemes for computing discontinuous solutions of the Degasperis-Procesi equation. Preprint, available at http://www.math.vio.no/~kennethk/articles/art125.pdf, 2006Google Scholar
  7. 7.
    Constantin A. (2000) Global existence of solutions and breaking waves for a shallow water equation: a geometric approach. Ann. Inst. Fourier (Grenoble) 50, 321–362MATHMathSciNetGoogle Scholar
  8. 8.
    Constantin A.(2005) Finite propagation speed for the Camassa-Holm equation. J. Math. Phys. 46(023506): 4MathSciNetGoogle Scholar
  9. 9.
    Constantin A. (2001) On the scattering problem for the Camassa-Holm equation. Proc. Roy. Soc. London A 457, 953–970MATHADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Constantin A., Escher J. (1998) Global existence and blow-up for a shallow water equation. Annali Sc. Norm. Sup. Pisa 26, 303–328MATHMathSciNetGoogle Scholar
  11. 11.
    Constantin A., Escher J. (1998) Wave breaking for nonlinear nonlocal shallow water equations. Acta Mathematica 181, 229–243MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Constantin A., Escher J. (1998) Global weak solutions for a shallow water equation. Indiana Univ. Math. J. 47, 1527–1545MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Constantin A., Kolev B. (2003) Geodesic flow on the diffeomorphism group of the circle. Comment. Math. Helv. 78, 787–804MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Constantin A., McKean H.P. (1999) A shallow water equation on the circle. Comm. Pure Appl. Math. 52, 949–982CrossRefMathSciNetGoogle Scholar
  15. 15.
    Constantin A., Molinet L. (2000) Global weak solutions for a shallow water equation. Commun. Math. Phys. 211, 45–61MATHCrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Constantin A., Strauss W.A. (2000) Stability of peakons. Comm. Pure Appl. Math. 53, 603–610MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Constantin A., Strauss W. (2000) Stability of a class of solitary waves in compressible elastic rods. Phys. Lett. A 270, 140–148MATHCrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Constantin A., Strauss W.A. (2002) Stability of the Camassa-Holm solitons. J. Nonlinear Science 12, 415–422MATHCrossRefADSMathSciNetGoogle Scholar
  19. 19.
    Dai H.H. (1998) Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod. Acta Mechanica 127, 193–207MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Degasperis A., Holm D.D., Hone A.N.W. (2002) A New Integral Equation with Peakon Solutions. Theo. Math. Phys. 133, 1463–1474CrossRefMathSciNetGoogle Scholar
  21. 21.
    Degasperis A., Procesi M.: Asymptotic integrability. In: Symmetry and Perturbation Theory, edited by A. Degasperis G. Gaeta, Singapore: World Scientific, 1999, pp. 23–37Google Scholar
  22. 22.
    Drazin P.G., Johnson R.S., (1989) Solitons: an Introduction. Cambridge-New York, Cambridge University PressMATHGoogle Scholar
  23. 23.
    Dullin H.R., Gottwald G.A., Holm D.D. (2001) An integrable shallow water equation with linear and nonlinear dispersion. Phys. Rev. Lett. 87, 4501–4504CrossRefADSGoogle Scholar
  24. 24.
    Dullin H.R., Gottwald G.A., Holm D.D. (2003) Camassa-Holm, Korteweg -de Vries-5 and other asymptotically equivalent equations for shallow water waves. Fluid Dyn. Res. 33, 73–79MATHCrossRefADSMathSciNetGoogle Scholar
  25. 25.
    Escher J., Liu Y., Yin Z.: Global weak solutions and blow-up structure for the Degasperis-Procesi equation. J. Funct. Anal., to appear, doi:10.1016/j.jfa.2006.03.022Google Scholar
  26. 26.
    Fokas A., Fuchssteiner B. (1981) Symplectic structures, their Bäcklund transformation and hereditary symmetries. Physica D 4, 47–66CrossRefADSMathSciNetGoogle Scholar
  27. 27.
    Henry D. (2005) Infinite propagation speed for the Degasperis-Procesi equation. J. Math. Anal. Appl. 311, 755–759MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Holm D.D., Staley M.F. (2003) Wave structure and nonlinear balances in a family of evolutionary PDEs. SIAM J. Appl. Dyn. Syst. (electronic) 2, 323–380MATHCrossRefMathSciNetADSGoogle Scholar
  29. 29.
    Johnson R.S. (2002) Camassa-Holm, Korteweg-de Vries and related models for water waves. J. Fluid Mech. 455, 63–82MATHCrossRefADSMathSciNetGoogle Scholar
  30. 30.
    T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations. In: Spectral Theory and Differential Equations, Lecture Notes in Math. 448 Berlin:Springer Verlag, 1975, pp. 25–70Google Scholar
  31. 31.
    Kenig C., Ponce G., Vega L. (1993) Well-posedness and scattering results for the generalized Korteweg-de Veris equation via the contraction principle. Comm. Pure Appl. Math. 46, 527–620MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Lenells J. (2005) Traveling wave solutions of the Degasperis-Procesi equation. J. Math. Anal. Appl. 306, 72–82MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Lenells J. (2005) Conservation laws of the Camassa-Holm equation. J. Phys. A 38, 869–880MATHCrossRefADSMathSciNetGoogle Scholar
  34. 34.
    Li P., Olver P. (2000) Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Differ. Eqs. 162, 27–63MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Liu Y. (2006) Global existence and blow-up solutions for a nonlinear shallow water equation. Math. Ann. 335, 717–735MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Lundmark H.: Formation and dynamics of shock waves in the Degasperis-Procesi equation. Preprint, available at http://www.mai.liu.se/~halun/papers/Lundmark-DPshock.pdf, 2006Google Scholar
  37. 37.
    Lundmark H., Szmigielski J. (2003) Multi-peakon solutions of the Degasperis-Procesi equation. Inverse Problems 19, 1241–1245MATHCrossRefADSMathSciNetGoogle Scholar
  38. 38.
    Matsuno Y. (2005) Multisoliton solutions of the Degasperis-Procesi equation and their peakon limit. Inverse Problems 21, 1553–1570MATHCrossRefADSMathSciNetGoogle Scholar
  39. 39.
    Mckean H.P.: Integrable systems and algebraic curves. In: Global Analysis. Springer Lecture Notes in Mathematics 755, Berlin-Heidelberg-New York:Springer, 1979, pp. 83–200Google Scholar
  40. 40.
    Misiolek G. (1998) A shallow water equation as a geodesic flow on the Bott-Virasoro group. J. Geom. Phys. 24, 203–208MATHCrossRefADSMathSciNetGoogle Scholar
  41. 41.
    Mustafa O.G. (2005) A note on the Degasperis-Procesi equation. J. Nonlinear Math. Phys. 12, 10–14MATHCrossRefMathSciNetADSGoogle Scholar
  42. 42.
    Rodriguez-Blanco G. (2001) On the Cauchy problem for the Camassa-Holm equation. Nonlinear Anal. 46, 309–327MATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Tao T.: Low-regularity global solutions to nonlinear dispersive equations. In: Surveys in analysis and operator theory (Canberra,2001), Proc. Centre Math. Appl. Austral. Nat. Univ. 40, Canberra:Austral. Nat. Univ., 2002, pp. 19–48Google Scholar
  44. 44.
    Vakhnenko V.O., Parkes E.J. (2004) Periodic and solitary-wave solutions of the Degasperis-Procesi equation. Chaos Solitons Fractals 20, 1059–1073MATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Whitham G.B., (1980) Linear and Nonlinear Waves. New York, J. Wiley & SonsGoogle Scholar
  46. 46.
    Xin Z., Zhang P. (2000) On the weak solutions to a shallow water equation. Comm. Pure Appl. Math. 53, 1411–1433MATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    Yin Z. (2003) On the Cauchy problem for an integrable equation with peakon solutions. Ill. J. Math. 47, 649–666MATHGoogle Scholar
  48. 48.
    Yin Z. (2003) Global existence for a new periodic integrable equation. J. Math. Anal. Appl. 283, 129–139MATHCrossRefMathSciNetGoogle Scholar
  49. 49.
    Yin Z. (2004) Global weak solutions to a new periodic integrable equation with peakon solutions. J. Funct. Anal. 212, 182–194MATHCrossRefMathSciNetGoogle Scholar
  50. 50.
    Yin Z.: Global solutions to a new integrable equation with peakons. Ind. Univ. Math. J. 53 (2004), 1189–1210 (2004)Google Scholar
  51. 51.
    Zhou Y. (2004) Blow-up phenomena for the integrable Degasperis-Procesi equation. Phys. Lett. A 328, 157–162MATHCrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TexasArlingtonUSA
  2. 2.Department of MathematicsZhongshan UniversityGuangzhouChina
  3. 3.Institute for Applied MathematicsUniversity of HanoverHanoverGermany

Personalised recommendations