A Liouville Property with Application to Asymptotic Stability for the Camassa–Holm Equation

Article
  • 21 Downloads

Abstract

We prove a Liouville property for uniformly almost localized (up to translations) H1-global solutions of the Camassa–Holm equation with a momentum density that is a non-negative finite measure. More precisely, we show that such a solution has to be a peakon. As a consequence, we prove that peakons are asymptotically stable in the class of H1-functions with a momentum density that belongs to \({\mathcal{M}_+(\mathbb{R})}\). Finally, we also get an asymptotic stability result for a train of peakons.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alvarez-Samaniego, B., Lannes, D.: Large time existence for 3D water-waves and asymptotics. Invent. Math. 171, 165–186 (2009)MathSciNetMATHGoogle Scholar
  2. 2.
    Beals, R., Sattinger, D.H., Szmigielski, J.: Multi-peakons and the classical moment problem. Adv. Math. 154(2), 229–257 (2000)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Benjamin, T.B.: The stability of solitary waves. Proc. R. Soc. Lond. Ser. A 328, 153–183 (1972)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Bourbaki, N.: Eléments de Mathématique, Intégration, Chapitre 9, Herman Paris 1969Google Scholar
  5. 5.
    Bressan, A., Chen, G., Zhang, Q.: Uniqueness of conservative solutions to the Camassa-Holm equation via characteristics. Discr. Contin. Dyn. Syst. 35, 25–42 (2015)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bressan, A., Constantin, A.: Global conservative solutions of the Camassa-Holm equation. Arch. Ration. Mech. Anal. 187, 215–239 (2007)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bressan, A., Constantin, A.: Global dissipative solutions of the Camassa-Holm equation. Anal. Appl. 5, 1–27 (2007)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cai, H., Chen, G., Chen, R.M., Shen, Y.: Lipschitz metric for the Novikov equation, arXiv:1611.08277
  9. 9.
    Camassa, R., Holm, D.: An integrable shallow water equation with peaked solitons. Phys. rev. Lett. 71, 1661–1664 (1993)ADSMathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Camassa, R., Holm, D., Hyman, J.: An new integrable shallow water equation. Adv. Appl. Mech. 31, 1994Google Scholar
  11. 11.
    Constantin, A.: Existence of permanent and breaking waves for a shallow water equations: a geometric approach. Ann. Inst. Fourier 50, 321–362 (2000)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Constantin, A.: On the scattering problem for the Camassa-Holm equation. Proc. R. Soc. Lond. Ser. A. 457, 953–970 (2001)ADSMathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Constantin, A., Escher, J.: Global existence and blow-up for a shallow water equation. Annali Sc. Norm. Sup. Pisa 26, 303–328 (1998)MathSciNetMATHGoogle Scholar
  14. 14.
    Constantin, A., Gerdjikov, V., Ivanov, R.: Inverse scattering transform for the Camassa-Holm equation. Inverse Probl. 22, 2197–2207 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Constantin, A., Lannes, D.: The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations Arch. Ration. Mech. Anal. 192, 165–186 (2009)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Constantin, A., Kolev, B.: Geodesic flow on the diffeomorphism group of the circle. Comment. Math. Helv. 78, 787–804 (2003)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Constantin, A., Molinet, L.: Global weak solutions for a shallow water equation. Commun. Math. Phys. 211, 45–61 (2000)ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Constantin, A., Strauss, W.: Stability of peakons. Commun. Pure Appl. Math. 53, 603–610 (2000)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Eckhardt, J., Teschl, G.: On the isospectral problem of the dispersionless Camassa-Holm equation. Adv. Math. 235, 469–495 (2013)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    El Dika, K., Martel, Y.: Stability of \(N\) solitary waves for the generalized BBM equations. Dyn. Partial Differ. Equ. 1, 401–437 (2004)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    El Dika, K., Molinet, L.: Stability of multipeakons. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(4), 1517–1532 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    El Dika, K., Molinet, L.: Stability of train of anti-peakons. Discrete Contin. Dyn. Syst. Ser. B 12(3), 561–577 (2009)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Grillakis, M., Shatah, J., Strauss, W.: Stability theory of solitary waves in the presence of symmetry. J. Funct. Anal. 74, 160–197 (1987)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Iftimie, D.: Large time behavior in perfect incompressible flows. Partial differential equations and applications, 119–179, Sémin. Congr., 15, Soc. Math. France, Paris, 2007Google Scholar
  25. 25.
    Johnson, R.S.: Camassa-Holm, Korteweg-de Vries and related models for water waves. J. Fluid Mech. 455, 63–82 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Kolev, B.: Lie groups and mechanics: an introduction. J. Nonlinear Math. Phys. 11, 480–498 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Kolev, B.: Poisson brackets in hydrodynamics. Discrete Contin. Dyn. Syst. 19, 555–574 (2007)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Martel, Y., Merle, F., Tsai, T.: Stability and asymptotic stability in the energy space of the sum of \(N\) solitons for subcritical gKdV equations. Commun. Math. Phys. 231, 347–373 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Martel, Y., Merle, F.: Asymptotic stability of solitons for subcritical generalized KdV equations. Arch. Ration. Mech. Anal. 157(3), 219–254 (2001)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Martel, Y., Merle, F.: Asymptotic stability of solitons of the gKdV equations with general nonlinearity. Math. Ann. 341(2), 391–427 (2008)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Molinet, L.: On well-posedness results for Camassa-Holm equation on the line: a survey. J. Nonlinear Math. Phys. 11, 521–533 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut Denis PoissonUniversité de Tours, Université d’Orléans, CNRSToursFrance

Personalised recommendations