Skip to main content
SpringerLink
Log in

Long time behavior of solutions for a damped Benjamin–Ono equation

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

We consider the Benjamin–Ono equation on the torus with an additional damping term on the smallest Fourier modes (\(\cos \) and \(\sin \)). We first prove global well-posedness of this equation in \(L^2_{r,0}(\mathbb {T})\). Then, we describe the weak limit points of the trajectories in \(L^2_{r,0}(\mathbb {T})\) when time goes to infinity, and show that these weak limit points are strong limit points. Finally, we prove the boundedness of higher-order Sobolev norms for this equation. Our key tool is the Birkhoff map for the Benjamin–Ono equation, that we use as an adapted nonlinear Fourier transform.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alarcon, E.A.: Existence and finite dimensionality of the global attractor for a class of nonlinear dissipative equations. Proc. R. Soc. Edinb. Section A Math. 123(5), 893–916 (1993)

    Article  MathSciNet  Google Scholar 

  2. Alazard, T., Bresch, D.: Functional inequalities and strong Lyapunov functionals for free surface flows in fluid dynamics (2020)

  3. Alazard, T., Meunier, N., Smets, D.: Lyapounov functions, identities and the Cauchy problem for the Hele-Shaw equation. Commun. Math. Phys. 377, 1421–1459 (2020)

    Article  Google Scholar 

  4. Baldi, P.: Periodic solutions of fully nonlinear autonomous equations of Benjamin-Ono type. Annales de l’IHP Analyse non linéaire 30(1), 33–77 (2013)

  5. Benjamin, T.B.: Internal waves of permanent form in fluids of great depth. J. Fluid Mech. 29(3), 559–592 (1967)

    Article  Google Scholar 

  6. Bernier, J., Grébert, B.: Long time dynamics for generalized Korteweg-de Vries and Benjamin-Ono equations (2020)

  7. Berti, M.: KAM theory for partial differential equations. Anal. Theory Appl. 35(3), 235–267 (2019)

    Article  MathSciNet  Google Scholar 

  8. Bock, T., Kruskal, M.: A two-parameter Miura transformation of the Benjamin-Ono equation. Phys. Lett. A 74(3–4), 173–176 (1979)

    Article  MathSciNet  Google Scholar 

  9. Boling, G., Yonghui, W.: Remarks on the global attractor of the weakly dissipative Benjamin-Ono equation. Northeast. Math. J. 11(4), 489–496 (1995)

    MathSciNet  MATH  Google Scholar 

  10. Bona, J.L., Luo, L.: Large-time asymptotics of the generalized Benjamin-Ono-Burgers equation. Discret. Continu. Dyn. Syst. S 4(1), 15 (2011)

    Article  MathSciNet  Google Scholar 

  11. Burq, N., Raugel, G., Schlag, W.: Long time dynamics for damped Klein-Gordon equations. Annales scientifiques de l’École Normale Supérieure 50(6), 1447–1498 (2017)

  12. Burq, N., Raugel, G., Schlag, W.: Long time dynamics for weakly damped nonlinear Klein-Gordon equations (2018)

  13. Coifman, R.R., Wickerhauser, M.V.: The scattering transform for the Benjamin-Ono equation. Inverse Prob. 6(5), 825 (1990)

    Article  MathSciNet  Google Scholar 

  14. Craig, W.: Problèmes de petits diviseurs dans les équations aux derivées partielles. Panoramas et syntheses, vol. 9, pp. 1–20 (2000)

  15. Deng, Y.: Invariance of the Gibbs measure for the Benjamin-Ono equation. J. Eur. Math. Soc. 17(5), 1107–1198 (2015)

    Article  MathSciNet  Google Scholar 

  16. Deng, Y., Tzvetkov, N., Visciglia, N.: Invariant measures and long time behaviour for the Benjamin-Ono equation III. Commun. Math. Phys. 339(3), 815–857 (2015)

    Article  MathSciNet  Google Scholar 

  17. Dix, D.B.: Temporal asymptotic behavior of solutions of the Benjamin-Ono-Burgers equation. J. Differ. Equ. 90(2), 238–287 (1991)

    Article  MathSciNet  Google Scholar 

  18. Dix, D.B.: The dissipation of nonlinear dispersive waves: the case of asymptotically weak nonlinearity. Commun. Partial Differ. Equ. 17(9–10), 1665–1693 (1992)

    Article  MathSciNet  Google Scholar 

  19. Fokas, A., Ablowitz, M.: The inverse scattering transform for the Benjamin-Ono equation—a pivot to multidimensional problems. Stud. Appl. Math. 68(1), 1–10, 2 (1983)

    Article  MathSciNet  Google Scholar 

  20. Gérard, P.: A nonlinear Fourier transform for the Benjamin–Ono equation on the torus and applications. Séminaire Laurent Schwartz—EDP et applications, pp. 1–19, (2019–2020)

  21. Gérard, P., Grellier, S.: The cubic Szegő equation and Hankel operators. (Astérisque, Société mathématique de France, Paris, 2017), pp 389

  22. Gérard, P., Grellier, S.: On a damped Szegő equation (with an appendix in collaboration With Christian Klein). SIAM J. Math. Anal. 52(5), 4391–4420 (2020)

    Article  MathSciNet  Google Scholar 

  23. Gérard, P., Kappeler, T.: On the integrability of the Benjamin-Ono equation on the torus. Commun. Pure Appl. Math. 74, 1685–1747 (2020)

    Article  MathSciNet  Google Scholar 

  24. Gérard, P., Kappeler, T., Topalov, P.: On the spectrum of the Lax operator of the Benjamin-Ono equation on the torus. J. Funct. Anal. 279(12), 108762 (2020)

    Article  MathSciNet  Google Scholar 

  25. Gérard, P., Kappeler, T., Topalov, P.: Sharp well-posedness results of the Benjamin-Ono equation in \(H^s(\mathbb{T},\mathbb{R})\) and qualitative properties of its solution (2020)

  26. Gérard, P., Kappeler, T., Topalov, P.: On the analytic Birkhoff normal form of the Benjamin-Ono equation and applications (2021)

  27. Grimshaw, R.: Internal solitary waves. In: Environmental stratified flows, pp. 1–17. Springer, New York (2003)

    Chapter  Google Scholar 

  28. Grimshaw, R.H., Smyth, N.F., Stepanyants, Y.A.: Decay of Benjamin-Ono solitons under the influence of dissipation. Wave Motion 78, 98–115 (2018)

    Article  MathSciNet  Google Scholar 

  29. Guo, B., Huo, Z.: The global attractor of the damped, forced generalized Korteweg de Vries-Benjamin-Ono equation in \(L^2\). Discret. Contin. Dyn. Syst. A 16(1), 121 (2006)

    Article  Google Scholar 

  30. Ifrim, M., Tataru, D.: Well-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation. Annales scientifiques de l’École Normale Supérieure 52(2), 297–335 (2019)

  31. Klein, C.: Private communication (2020)

  32. Kuksin, S.B.: Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum. Funct. Anal. Appl. 21(3), 192–205 (1987)

    Article  MathSciNet  Google Scholar 

  33. Liu, J., Yuan, X.: A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations. Commun. Math. Phys. 307(3), 629 (2011)

    Article  MathSciNet  Google Scholar 

  34. Matsuno, Y., Shchesnovich, V.S., Kamchatnov, A.M., Kraenkel, R.A.: Whitham method for the Benjamin-Ono-Burgers equation and dispersive shocks. Phys. Rev. E 75, 016307, 1 (2007)

    Article  MathSciNet  Google Scholar 

  35. Mi, L., Zhang, K.: Invariant tori for Benjamin-Ono equation with unbounded quasi-periodically forced perturbation. Discret. Contin. Dyn. Syst. A 34(2), 689–707 (2014)

    Article  MathSciNet  Google Scholar 

  36. Molinet, L.: Global well-posedness in \(L^2\) for the periodic Benjamin-Ono equation. Am. J. Math. 130(3), 635–683 (2008)

    Article  Google Scholar 

  37. Molinet, L., Pilod, D.: The Cauchy problem for the Benjamin-Ono equation in \(L^2\) revisited. Anal. PDE 5(2), 365–395 (2012)

    Article  MathSciNet  Google Scholar 

  38. Nakamura, A.: Bäcklund transform and conservation laws of the Benjamin-Ono equation. J. Phys. Soc. Jpn. 47(4), 1335–1340 (1979)

    Article  Google Scholar 

  39. Ono, H.: Algebraic solitary waves in stratified fluids. J. Phys. Soc. Jpn. 39(4), 1082–1091 (1975)

    Article  MathSciNet  Google Scholar 

  40. Ott, E., Sudan, R.: Damping of solitary waves. Phys. Fluids 13(6), 1432–1434 (1970)

    Article  Google Scholar 

  41. Saut, J.-C.: Benjamin-Ono and intermediate long wave equations: modeling. In: Nonlinear Dispersive Partial Differential Equations and Inverse Scattering. IST and PDE, pp. 95–160. Springer, New York (2019)

    Chapter  Google Scholar 

  42. Sun, R.: Complete integrability of the Benjamin-Ono equation on the multi-soliton manifolds. Commun. Math. Phys. 383, 1051–1092, 4 (2021)

    Article  MathSciNet  Google Scholar 

  43. Sy, M.: Invariant measure and long time behavior of regular solutions of the Benjamin-Ono equation. Anal. PDE 11(8), 1841–1879 (2018)

    Article  MathSciNet  Google Scholar 

  44. Tzvetkov, N.: Construction of a Gibbs measure associated to the periodic Benjamin-Ono equation. Probab. Theory Related Fields 146(3–4), 481, 3 (2010)

    MathSciNet  MATH  Google Scholar 

  45. Tzvetkov, N., Visciglia, N.: Gaussian measures associated to the higher order conservation laws of the Benjamin-Ono equation. Annales scientifiques de l’École Normale Supérieure 46(2), 249–299 (2013)

  46. Tzvetkov, N., Visciglia, N.: Invariant measures and long-time behavior for the Benjamin-Ono equation. Int. Math. Res. Notices 2014(17), 4679–4714, 05 (2013)

    Article  MathSciNet  Google Scholar 

  47. Tzvetkov, N., Visciglia, N.: Invariant measures and long time behaviour for the Benjamin-Ono equation II. Journal de Mathématiques Pures et Appliquées 103(1), 102–141 (2015)

    Article  MathSciNet  Google Scholar 

  48. Wayne, C.E.: Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory. Commun. Math. Phys. 127(3), 479–528 (1990)

    Article  MathSciNet  Google Scholar 

  49. Wu, Y.: Simplicity and finiteness of discrete spectrum of the Benjamin-Ono scattering operator. SIAM J. Math. Anal. 48(2), 1348–1367 (2016)

    Article  MathSciNet  Google Scholar 

  50. Wu, Y.: Jost Solutions and the direct scattering problem of the Benjamin-Ono equation. SIAM J. Math. Anal. 49(6), 5158–5206 (2017)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author would like to thank her PhD advisor Patrick Gérard who introduced her to this problem and provided generous advice and encouragement. She also warmly thanks Christian Klein for valuable discussions and numerical simulations.

Author information

Authors and Affiliations

  1. Département de mathématiques et applications, École normale supérieure, CNRS, PSL University, 75005, Paris, France

    Louise Gassot

  2. Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay, 91405, Orsay, France

    Louise Gassot

Authors
  1. Louise Gassot
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Louise Gassot.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gassot, L. Long time behavior of solutions for a damped Benjamin–Ono equation. Math. Z. 300, 1939–2006 (2022). https://doi.org/10.1007/s00209-021-02849-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-021-02849-w

Keywords

Mathematics Subject Classification

Search

Navigation