Advertisement

On the Cauchy problem for the new integrable two-component Novikov equation

  • Yongsheng MiEmail author
  • Chunlai Mu
Article
  • 20 Downloads

Abstract

This paper is devoted to a new integrable two-component Novikov equation, lax pairs and bi-Hamiltonian structures. Firstly, the local well-posedness in nonhomogeneous Besov spaces is established by using the Littlewood–Paley theory and transport equations theory. Then, we verify the blow-up that occurs for this system only in the form of breaking waves. Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. Finally, we prove that the strong solutions of the system maintain corresponding properties at infinity within its lifespan provided the initial data decay exponentially and algebraically, respectively.

Keywords

Besov spaces Camassa–Holm-type equation Local well-posedness Persistence properties 

Mathematics Subject Classification

35B30 35G25 35A10 35Q53 

Notes

Acknowledgements

The authors would like to thank the referees for constructive suggestions and comments. The work of Mi is partially supported by NSF of China (11671055), partially supported by NSF of Chongqing (cstc2018jcyjAX0273), partially supported by Key project of science and technology research program of Chongqing Education Commission (KJZD-K20180140).

References

  1. 1.
    Baouendi, M., Goulaouic, C.: Sharp estimates for analytic pseudodifferential operators and application to the Cauchy problems. J. Differ. Equ. 48, 241–268 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bona, J., Smith, R.: The initial-value problem for the Korteweg-de Vries equation. Philos. Trans. R. Soc. Lond. Ser. A 278, 555–601 (1975)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.
    Bressan, A., Constantin, A.: Global conservative solutions of the Camassa–Holm equation. Arch. Ration. Mech. Anal. 183, 215–239 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Chemin, J.: Localization in Fourier space and Navier–Stokes system, Phase Space Analysis of Partial Differential Equations. In: Proceedings, CRM series, Pisa, 53–136 (2004)Google Scholar
  6. 6.
    Constantin, A.: Finite propagation speed for the Camassa–Holm equation. J. Math. Phys. 46, 023506 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Constantin, A.: On the scattering problem for the Camassa–Holm equation. Proc. R. Soc. Lond. A 457, 953–970 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Constantin, A.: The trajectories of particles in Stokes waves. Invent. Math. 166, 23–535 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Constantin, A.: On the inverse spectral problem for the Camassa–Holm equation. J. Funct. Anal. 155, 352–363 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Constantin, A., Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181, 229–243 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Constantin, A., Escher, J.: Particle trajectories in solitary water waves. Bull. Am. Math. Soc. 44, 423–431 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Constantin, A., Escher, J.: Analyticity of periodic traveling free surface water waves with vorticity. Ann. Math. 173, 559–568 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Constantin, A., Gerdjikov, V., Ivanov, R.I.: Inverse scattering transform for the Camassa–Holm equation. Inverse Probl. 22, 2197–2207 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Constantin, A., Kappeler, T., Kolev, B., Topalov, T.: On Geodesic exponential maps of the Virasoro group. Ann. Global Anal. Geom. 31, 155–180 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Constantin, A., Lannes, D.: The hydro-dynamical relevance of the Camassa–Holm and Degasperis–Procesi equations. Arch. Ration. Mech. Anal. 193, 165–186 (2009)zbMATHCrossRefGoogle Scholar
  16. 16.
    Constantin, A., McKean, H.P.: A shallow water equation on the circle. Commun. Pure Appl. Math. 52, 949–982 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Camassa, R., Holm, D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Constantin, A., Ivanov, R., Lenells, J.: Inverse scattering transform for the Degasperis–Procesi equation. Nonlinearity 23, 2559–2575 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Constantin, A., Strauss, W.: Stability of peakons. Commun. Pure Appl. Math. 53, 603–610 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Danchin, R.: A few remarks on the Camassa–Holm equation. Differ. Integral Equ. 14, 953–988 (2001)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Danchin, R.: Fourier analysis methods for PDEs, Lecture Notes, 14 (2003)Google Scholar
  22. 22.
    Degasperis, A., Holm, D.D., Hone, A.N.W.: A new integrable equation with peakon solutions. Theor. Math. Phys. 133, 1461–72 (2002)MathSciNetGoogle Scholar
  23. 23.
    Degasperis, A., Procesi, M.: Asymptotic integrability. In: Degasperis, A., Gaeta, G. (eds.) Symmetry and Perturbation Theory, pp. 23–37. World Scientific, Singapore (1999)Google Scholar
  24. 24.
    Fokas, A., Fuchssteiner, B.: Symplectic structures, their Backlund transformation and hereditary symmetries. Physica D 4, 47–66 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Geng, X., Xue, B.: An extension of integrable peakon equations with cubic nonlinearity. Nonlinearity 22, 1847–1856 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Gui, G., Liu, Y.: On the global existence and wave-breaking criteria for the two-component Camassa–Holm system. J. Funct. Anal. 258, 4251–4278 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Grillakis, M., Shatah, J., Strauss, W.: Stability theory of solitary waves in the presence of symmetry. J. Funct. Anal. 74, 160–197 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Himonas, A., Mantzavinos, D.: The initial value problem for a Novikov system. J. Math. Phys. 57, 071503 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Henry, D.: Compactly supported solutions of the Camassa–Holm equation. J. Nonlinear Math. Phys. 12, 342–347 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Henry, D.: Persistence properties for a family of nonlinear partial differential equations. Nonlinear Anal. 70, 1565–1573 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Himonas, A., Misiolek, G.: Analyticity of the Cauchy problem for an integrable evolution equation. Math. Ann. 327, 575–584 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Himonas, A., Holliman, C.: The Cauchy problem for the Novikov equation. Nonlinearity 25, 449–479 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Himonas, A., Holliman, C.: The Cauchy problem for a generalized Camassa–Holm equation. Adv. Differ. Equ. 19, 161–200 (2014)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Himonas, A., Misiołek, G.: Non-uniform dependence on initial data of solutions to the Euler equations of hydrodynamics. Commun. Math. Phys. 296, 285–301 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Himonas, A., Misiołlek, G., Ponce, G., Zhou, Y.: Persistence properties and unique continuation of solutions of the Camassa–Holm equation. Commun. Math. Phys. 271, 511–522 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Holden, H., Raynaud, X.: Dissipative solutions for the Camassa–Holm equation. Discret. Contin. Dyn. Syst. 24, 1047–1112 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Holden, H., Raynaud, X.: Global conservative solutions of the Camassa–Holm equations-a Lagrangianpoiny of view. Commun. Partial Differ. Equ. 32, 1511–1549 (2007)zbMATHCrossRefGoogle Scholar
  38. 38.
    Holm, D.D., Staley, M.F.: Wave structure and nonlinear balances in a family of evolutionary PDEs. SIAM J. Appl. Dyn. Syst. 2, 323–380 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Home, A.N.W., Wang, J.P.: Integrable peakon equations with cubic nonlinearity. J. Phys. A 41, 372002 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Hone, W., Lundmark, H., Szmigielski, J.: Explicit multipeakon solutions of Novikov cubically nonlinear integrable Camassa–Holm type equation. Dyn. Partial Differ. Equ. 6, 253–289 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Hu, Q., Qiao, Z: Analyticity, Gevrey regularity and unique continuation for an integrable multi-component peakon system with an arbitrary polynomial function. arXiv:1511.03315 (2015)
  42. 42.
    Jiang, Z., Ni, L.: Blow-up phenomenon for the integrable Novikov equation. J. Math. Anal. Appl. 385, 551–558 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier–Stokes equations. Commun. Pure Appl. Math. 41, 891–907 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Lai, S., Wu, Y.: The local well-posedness and existence of weak solutions for a generalized Camassa–Holm equation. J. Differ. Equ. 248, 2038–2063 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Lenells, J.: Traveling wave solutions of the Degasperis–Procesi equation. J. Math. Anal. Appl. 306, 72–82 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Li, H.: Two-component generalizations of the Novikov equation. J. Nonlinear Math. Phys. 26, 390–403 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Liu, X., Yin, Z.: Local well-posedness and stability of peakons for a generalized Dullin–Gottwald–Holm equation. Nonlinear Anal. 74, 2497–2507 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Li, H., Liu, Q.P.: On bi-Hamiltonian structure of two-component Novikov equation. Phys. Lett. A 377, 257–281 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Lundmark, H., Szmigielski, J.: Multi-peakon solutions of the Degasperis–Procesi equation. Inverse Probl. 21, 1553–1570 (2005)zbMATHCrossRefGoogle Scholar
  50. 50.
    Matsuno, Y.: Multisoliton solutions of the Degasperis–Procesi equation and their peakon limit. Inverse Probl. 19, 1241–1245 (2003)CrossRefGoogle Scholar
  51. 51.
    Misiolek, G.A.: Shallow water equation as a geodesic flow on the Bott–Virasoro group. J. Geom. Phys. 24, 203–208 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Ni, L., Zhou, Y.: Well-posedness and persistence properties for the Novikov equation. J. Differ. Equ. 250, 3002–3021 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Ni, L., Zhou, Y.: A new asymptotic behavior of solutions to the Camassa–Holm equation. Proc. Am. Math. Soc. 140, 607–614 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Novikov, V.: Generalization of the Camassa–Holm equation. J. Phys. A 42, 342002 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Taylor, M.: Partial Differential Equations III, Nonlinear Equations. Springer, Berlin (1996)CrossRefGoogle Scholar
  56. 56.
    Tiglay, F.: The periodic Cauchy problem for Novikov equation. Int. Math. Res. Not. 2011, 4633–4648 (2011)MathSciNetzbMATHGoogle Scholar
  57. 57.
    Toland, J.F.: Stokes waves. Topol Methods Nonlinear Anal. 7, 1–48 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Vakhnenko, V.O., Parkes, E.J.: Periodic and solitary-wave solutions of the Degasperis–Procesi equation. Chaos Solitons Fractals 20, 1059–1073 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Wu, X., Yin, Z.: Well-posedness and global existence for the Novikov equation. Annali Sc. Norm. Sup. Pisa. X I, 707–727 (2012)MathSciNetzbMATHGoogle Scholar
  60. 60.
    Wu, X., Yin, Z.: Global weak solutions for the Novikov equation. J. Phys. A 44, 055202 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    Xia, B., Qiao, Z.: A new two-component integrable system with peakon solutions. Proc. R. Soc. A Math. Phys. Eng. 471, 20140750 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    Xia, B., Qiao, Z., Zhou, R.: A synthetical two-component model with Peakon solutions. Stud. Appl. Math. 135, 248–276 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Yan, K., Qiao, Z., Yin, Z.: Qualitative analysis for a new integrable two-component Camassa–Holm system with peakon and weak kink solutions. Commun. Math. Phys. 336, 581–617 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    Yan, W., Li, Y., Zhang, Y.: Global existence and blow-up phenomena for the weakly dissipative Novikov equation. Nonlinear Anal. 75, 2464–2473 (2012)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.College of Mathematics and StatisticsYangtze Normal UniversityChongqingPeople’s Republic of China
  2. 2.College of Mathematics and StatisticsChongqing UniversityChongqingPeople’s Republic of China

Personalised recommendations