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On well-posedness for the Benjamin–Ono equation

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We prove existence and uniqueness of solutions for the Benjamin–Ono equation with data in \(H^{s}({\mathbb{R}})\) , s > 1/4. Moreover, the flow is hölder continuous in weaker topologies.

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References

  1. Albert J.P., Bona J.L. and Saut J.-C. (1997). Model equations for waves in stratified fluids. Proc. Roy. Soc. Lond. Ser. A 453(1961): 1233–1260

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  3. Bergh, J., Löfström, J.: Interpolation spaces. An introduction. Springer, Berlin, Grundlehren der Mathematischen Wissenschaften, No. 223 (1976)

  4. Burq, N., Planchon, F.: Smoothing and dispersive estimates for 1d Schrödinger equations with BV coefficients and applications. J. Funct. Anal. 236(1), 265–298 (2006)

    Google Scholar 

  5. Burq N., Gérard P. and Tzvetkov N. (2004). Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds. Am. J. Math. 126(3): 569–605

    Article  MATH  Google Scholar 

  6. Colliander, J.E., Delort, J.-M., Kenig, C.E., Staffilani, G.: Bilinear estimates and applications to 2D NLS. Trans. Am. Math. Soc. 353(8), 3307–3325 (electronic) (2001)

    Google Scholar 

  7. Ginibre J. and Velo G. (1991). Smoothing properties and existence of solutions for the generalized Benjamin-Ono equation. J. Differ. Equ. 93(1): 150–212

    Article  MATH  MathSciNet  Google Scholar 

  8. Hayashi N. and Ozawa T. (1994). Remarks on nonlinear Schrödinger equations in one space dimension. Differ. Integr. Equ. 7(2): 453–461

    MATH  MathSciNet  Google Scholar 

  9. Herr, S.: An improved bilinear estimate for Benjamin–Ono type equations. Preprint, arXiv:math.AP/ 0509218 (2005)

  10. Ionescu, A.D., Carlos, E.K.: Global well-posedness of the Benjamin-Ono equation in low regularity spaces. J. Amer. Math. Soc. 20(3), 753–798

  11. Ionescu, A.D., Carlos, E.K.: Complex-valued solutions of the Benjamin-Ono equation. Preprint, arXiv:math/0605158v1 (2006)

  12. Kappeler T. and Topalov P. (2006). Global wellposedness of KdV in \(H^{-1}({\mathbb{T}},{\mathbb{R}})\). Duke Math. J. 135(2): 327–360

    Article  MATH  MathSciNet  Google Scholar 

  13. Kenig C.E. and Koenig K.D. (2003). On the local well-posedness of the Benjamin-Ono and modified Benjamin-Ono equations. Math. Res. Lett. 10(5–6): 879–895

    MATH  MathSciNet  Google Scholar 

  14. Kenig C.E., Ponce G. and Vega L. (1996). Quadratic forms for the 1-D semilinear Schrödinger equation. Trans. Am. Math. Soc. 348(8): 3323–3353

    Article  MATH  MathSciNet  Google Scholar 

  15. Kenig C.E., Ponce G. and Vega L. (1991). Oscillatory integrals and regularity of dispersive equations. Indiana Univ. Math. J. 40(1): 33–69

    Article  MATH  MathSciNet  Google Scholar 

  16. Kenig C.E., Ponce G. and Vega L. (1994). On the generalized Benjamin-Ono equation. Trans. Am. Math. Soc. 342(1): 155–172

    Article  MATH  MathSciNet  Google Scholar 

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

  18. Koch, H., Tzvetkov, N.: Nonlinear wave interactions for the Benjamin-Ono equation. Int. Math. Res. Not. (30), 1833–1847 (2005)

  19. Molinet, L. Global well-posedness in L 2 for the periodic Benjamin-Ono equation. Preprint, arXiv:math.AP/0601217 (2006)

  20. Molinet, L.: Global well-posedness in the energy space for the Benjamin-Ono equation on the circle. Math. Ann. 337(2) 353–383 (2007)

    Google Scholar 

  21. Molinet, L., Saut, J.C., Tzvetkov, N.: Ill-posedness issues for the Benjamin-Ono and related equations. SIAM J. Math. Anal. 33(4), 982–988 (electronic) (2001)

    Google Scholar 

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

    Article  Google Scholar 

  23. Ponce G. (1991). On the global well-posedness of the Benjamin-Ono equation. Differ. Integr. Equ. 4(3): 527–542

    MATH  MathSciNet  Google Scholar 

  24. Tao T. (2001). Multilinear weighted convolution of L 2-functions and applications to nonlinear dispersive equations. Am. J. Math. 123(5): 839–908

    Article  MATH  Google Scholar 

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

    Google Scholar 

  26. Vega, L.: Restriction theorems and the Schrödinger multiplier on the torus. Partial differential equations with minimal smoothness and applications (Chicago, IL, 1990), pp. 199–211, IMA Vol. Math. Appl., 42, Springer, New York (1992)

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Correspondence to Fabrice Planchon.

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Burq, N., Planchon, F. On well-posedness for the Benjamin–Ono equation. Math. Ann. 340, 497–542 (2008). https://doi.org/10.1007/s00208-007-0150-y

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  • DOI: https://doi.org/10.1007/s00208-007-0150-y

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