Skip to main content
SpringerLink
Log in

Weak Continuity of the Flow Map for the Benjamin-Ono Equation on the Line

  • Published:
Journal of Fourier Analysis and Applications Aims and scope Submit manuscript

Abstract

In this paper we show that the flow map of the Benjamin-Ono equation on the line is weakly continuous in L 2(ℝ), using “local smoothing” estimates. L 2(ℝ) is believed to be a borderline space for the local well-posedness theory of this equation. In the periodic case, Molinet (Math. Ann. 337, 353–383, 2007) has recently proved that the flow map of the Benjamin-Ono equation is not weakly continuous in \(L^{2}(\mathbb{T})\). Our results are in line with previous work on the cubic nonlinear Schrödinger equation, where Goubet and Molinet (Nonlinear Anal. 71, 317–320, 2009) showed weak continuity in L 2(ℝ) and Molinet (Am. J. Math. 130, 635–683, 2008) showed lack of weak continuity in \(L^{2}(\mathbb{T})\).

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

    Article  MATH  Google Scholar 

  2. Biagioni, H.A., Linares, F.: Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations. Trans. Am. Math. Soc. 353, 3649–3659 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. Burq, N., Planchon, F.: On well-posedness for the Benjamin-Ono equation. Math. Ann. 340, 497–542 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Christ, M., Colliander, J., Tao, T.: Asymptotics, frequency modulation and low-regularity ill-posedness for canonical defocusing equations. Am. J. Math. 125, 1235–1293 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  5. Colliander, J., Kenig, C.E., Staffilani, G.: Local well-posedness for dispersion-generalized Benjamin-Ono equations. Differ. Integral Equ. 16, 1441–1472 (2003)

    MATH  MathSciNet  Google Scholar 

  6. Ginibre, J., Velo, G.: Properties de lissage et existence de solutions pour l’equation de Benjamin-Ono generalisee. C. R. Acad. Sci. Paris Ser. I. Math. 308, 309–314 (1989)

    MATH  MathSciNet  Google Scholar 

  7. Ginibre, J., Velo, G.: Commutator expansions and smoothing properties of generalized Benjamin-Ono equations. Ann. Inst. H. Poincaré, Phys. Theor. 51, 221–229 (1989)

    MATH  MathSciNet  Google Scholar 

  8. Ginibre, J., Velo, G.: Smoothing properties and existence of solutions for the generalized Benjamin-Ono equation. J. Differ. Equ. 93, 150–212 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  9. Goubet, O., Molinet, L.: Global weak attractor for weakly damped nonlinear Schrödinger equations in L 2(R). Nonlinear Anal. 71, 317–320 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  10. Folland, G.B.: Real Analysis, 2nd edn. Wiley, New York (1999)

    MATH  Google Scholar 

  11. Ionescu, A.D., Kenig, C.E.: Global well-posedness of the Benjamin-Ono equation in low-regularity spaces. J. Am. Math. Soc. 20, 753–798 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Iorio, R.J.: On the Cauchy problem for the Benjamin-Ono equation. Commun. Partial Differ. Equ. 11, 1031–1081 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kenig, C.E., Koenig, K.D.: On the local well-posedness of the Benjamin-Ono and modified Benjamin-Ono equations. Math. Res. Lett. 10, 879–895 (2003)

    MATH  MathSciNet  Google Scholar 

  14. Kenig, C.E., Martel, Y.: Asymptotic stability of solitons for the Benjamin-Ono equation. Rev. Mat. Iberoam. (to appear, see also arXiv:0803.3683)

  15. Kenig, C.E., Takaoka, H.: Global well-posedness of the modified Benjamin-Ono equation with initial data in H 1/2. Int. Math. Res. Not. 2006, 1–44 (2006)

    MathSciNet  Google Scholar 

  16. Kenig, C.E., Ponce, G., Vego, L.: Well-posedness and scattering results for the generalized Korteweg-de Vries equation via contraction principle. Commun. Pure Appl. Math. 46, 527–620 (1993)

    Article  MATH  Google Scholar 

  17. Kenig, C.E., Ponce, G., Vega, L.: On the generalized Benjamin-Ono equation. Trans. Am. Math. Soc. 342, 155–172 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kenig, C.E., Ponce, G., Vega, L.: On the ill-posedness of some canonical dispersive equations. Duke Math. J. 106, 617–633 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kenig, C.E., Martel, Y., Robbiano, L.: Local well-posedness and blow-up in the energy space for a class of L2 critical dispersive generalized Benjamin-Ono equations. arXiv:1006.0122

  20. Koch, H., Tzvetkov, N.: On the local well-posedness of the Benjamin-Ono equation in H s(R). Int. Math. Res. Not. 2003(26), 1449–1464 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  21. Martel, Y., Merle, F.: A Liouville theorem for the critical generalized Korteweg-de Vries equation. J. Math. Pures Appl. 79, 339–425 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  22. Martel, Y., Merle, F.: Instability of solitons for the critical generalized Korteweg-de Vries equation. Geom. Funct. Anal. 38, 759–781 (2001)

    MathSciNet  Google Scholar 

  23. Martel, Y., Merle, F.: Asymptotic stability of solitons for subcritical generalized KdV equations. Arch. Ration. Mech. Anal. 157, 219–254 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  24. Martel, Y., Merle, F.: Blow up in finite time and dynamics of blow up solutions for the L 2-critical generalized KdV equation. J. Am. Math. Soc. 15, 617–664 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  25. Molinet, L.: Global well-posedness in the energy space for the Benjamin-Ono equation on the circle. Math. Ann. 337, 353–383 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  26. Molinet, L.: Global well-posedness in L2 for the periodic Benjamin-Ono equation. Am. J. Math. 130, 635–683 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  27. Molinet, L.: On ill-posedness for the one-dimensional periodic cubic Schrödinger equation. Math. Res. Lett. 16, 111–120 (2009)

    MATH  MathSciNet  Google Scholar 

  28. Molinet, L.: Sharp ill-posedness result for the periodic Benjamin-Ono equation, arXiv:0811.0505

  29. Molinet, L., Ribaud, F.: Well-posedness results for the generalized Benjamin-Ono equation with small initial data. J. Math. Pures Appl. 83, 277–311 (2004)

    MATH  MathSciNet  Google Scholar 

  30. Molinet, L., Ribaud, F.: Well-posedness results for the generalized Benjamin-Ono equation with arbitrary large initial data. Int. Math. Res. Not. 2004(70), 3757–3795 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  32. Ponce, G.: On the global well-posedness of the Benjamin-Ono equation. Differ. Integral Equ. 4, 527–542 (1991)

    MATH  MathSciNet  Google Scholar 

  33. Tao, T.: Global well-posedness of the Benjamin-Ono equation in H 1(ℝ). J. Hyperbolic Differ. Equ. 1, 27–49 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  34. Tsutsumi, Y.: L 2-solutions for nonlinear Schrödinger equations and nonlinear groups. Funkc. Ekvacioj. 30, 115–125 (1987)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Sun Yat-Sen University, Guangzhou, Guangdong, 510275, People’s Republic of China

    Shangbin Cui

  2. Department of Mathematics, University of Chicago, Chicago, IL, 60637, USA

    Carlos E. Kenig

Authors
  1. Shangbin Cui
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Carlos E. Kenig
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shangbin Cui.

Additional information

Communicated by H. Feichtinger.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cui, S., Kenig, C.E. Weak Continuity of the Flow Map for the Benjamin-Ono Equation on the Line. J Fourier Anal Appl 16, 1021–1052 (2010). https://doi.org/10.1007/s00041-010-9137-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00041-010-9137-2

Keywords

Search

Navigation