Advertisement

The Cauchy problem for the Ostrovsky equation with positive dispersion

  • Wei Yan
  • Yongsheng Li
  • Jianhua Huang
  • Jinqiao Duan
Article
  • 44 Downloads

Abstract

This paper is devoted to studying the Cauchy problem for the Ostrovsky equation
$$\begin{aligned} \partial _{x}\left( u_{t}-\beta \partial _{x}^{3}u +\frac{1}{2}\partial _{x}(u^{2})\right) -\gamma u=0, \end{aligned}$$
with positive \(\beta \) and \(\gamma \). This equation describes the propagation of surface waves in a rotating oceanic flow. We first prove that the problem is locally well-posed in \(H^{-\frac{3}{4}}(\text{ R })\). Then we reestablish the bilinear estimate, by means of the Strichartz estimates instead of calculus inequalities and Cauchy–Schwartz inequalities. As a byproduct, this bilinear estimate leads to the proof of the local well-posedness of the problem in \(H^{s}(\text{ R })\) for \( s>-\frac{3}{4}\), with help of a fixed point argument.

Keywords

Ostrovsky equation with positive dispersion Cauchy problem Bilinear estimates Strichartz estimates 

Mathematics Subject Classification

Primary 35Q53 Secondary 35B30 

References

  1. 1.
    Beals, M.: Self-spreading and strength of singularities for solutions to semilinear wave equations. Ann. Math. 118, 187–214 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bejenaru, I., Tao, T.: Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation. J. Funct. Anal. 233, 228–259 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Benilov, E.S.: On the surface waves in a shallow channel with an uneven bottom. Stud. Appl. Math. 87, 1–14 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bourgain, J.: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part I: Schrödinger equations. Geom. Funct. Anal. 3, 107–156 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bourgain, J.: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part II: the KdV equation. Geom. Funct. Anal. 3, 209–262 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bourgain, J.: Periodic Korteweg de vries equation with measures as initial data. Sel. Math. 3, 115–159 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Boyd, J.: Ostrovsky and Hunter’s generic wave equation for weakly dispersive waves: Matched asymptotic and pseudospectral study of the paraboloidal travelling waves (corner and nearcorner waves). Eur. J. Appl. Math. 16, 65–81 (2005)CrossRefzbMATHGoogle Scholar
  8. 8.
    Buckmaster, T., Koch, H.: Herbert the Korteweg-de Vries equation at \(H^{-1}\) regularity. Ann. I. H. Poincare-AN 32, 1071–1098 (2015)CrossRefzbMATHGoogle Scholar
  9. 9.
    Choudhury, R., Ivanov, R., Liu, Y.: Hamiltonian formulation, nonintegrability and local bifurcations for the Ostrovsky equation. Chaos, Solitons Fractals 34, 544–550 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Christ, M., Colliander, J., Tao, T.: Asympotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations. Am. J. Math. 125, 1235–1293 (2003)CrossRefzbMATHGoogle Scholar
  11. 11.
    Coclite, G.M., di Ruvo, L., Karlsen, K.H.: Some wellposedness results for the Ostrovsky–Hunter equation, hyperbolic conservation laws and related analysis with applications, pp. 143–159. Springer Proc. Math. Stat. 49, Springer, Heidelberg (2014)Google Scholar
  12. 12.
    Colliander, J., Keel, M., Staffilani, G., Takaoka, H., Tao, T.: Sharp global well-posedness for KdV and modified KdV on \(R\) and \(T\). J. Am. Math. Soc. 16, 705–749 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Coclite, G.M., Di Ruvo, L.: Lorenzo, convergence of the Ostrovsky equation to the Ostrovsky–Hunter one. J. Differ. Equ. 256, 3245–3277 (2014)CrossRefzbMATHGoogle Scholar
  14. 14.
    Coclite, G.M., Di Ruvo, L.: Lorenzo, dispersive and diffusive limits for Ostrovsky–Hunter type equations. Nonl. Differ. Equ. Appl. 22, 1733–1763 (2015)CrossRefzbMATHGoogle Scholar
  15. 15.
    Galkin, V.N., Stepanyants, Y.A.: On the existence of stationary solitary waves in a rotating fluid. J. Appl. Math. Mech. 55, 939–943 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gilman, O.A., Grimshaw, R., Stepanyants, Y.A.: Approximate and numerical solutions of the stationary Ostrovsky equation. Stud. Appl. Math. 95, 115–126 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ginibre, J., Tsutsumi, Y., Velo, G.: On the Cauchy problem for the Zakharov system. J. Funct. Anal. 151, 384–436 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Grimshaw, R.: Evolution equations for weakly nonlinear long internal waves in a rotating fluid. Stud. Appl. Math. 73, 1–33 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Grünrock, A.: New applications of the Fourier restriction norm method to wellposedness problems for nonlinear Evolution Equations. Ph.D. Universit\(\ddot{a}\)t Wuppertal, 2002, Germany, Dissertation (2002)Google Scholar
  20. 20.
    Gui, G.L., Liu, Y.: On the Cauchy problem for the Ostrovsky equation with positive dispersion. Commun. Partial Differ. Equ. 32, 1895–1916 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Guo, B.L., Huo, Z.H.: The global attractor of the damped forced Ostrovsky equation. J. Math. Anal. Appl. 329, 392–407 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Guo, Z.H.: Global well-posedness of the Korteweg-de Vries equation in \(H^{-3/4}\). J. Math. Pures Appl. 91, 583–597 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hunter, J.: Numerical solutions of some nonlinear dispersive wave equations. In: Computational Solution of Nonlinear Systems of Equations, Lectures in Appl. Math. 26, AMS, Providence, RI, pp. 301–316 (1990)Google Scholar
  24. 24.
    Huo, Z.H., Jia, Y.L.: Low regularity solution for a nonlocal perturbation of KdV equation. Zeitschrift fü rangewandte Mathematik und Physik 59, 634–646 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Huo, Z., Jia, Y.L.: Low-regularity solutions for the Ostrovsky equation. Proc. Edinb. Math. Soc. 49, 87–100 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Ionescu, A.D., Kenig, C.E., Tataru, D.: Global well-posedness of the KP-I initial-value problem in the energy space. Invent. Math. 173, 265–304 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Isaza, P., Mejía, J.: Cauchy problem for the Ostrovsky equation in spaces of low regularity. J. Differ. Equ 230, 661–681 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Isaza, P., Mejía, J.: Global Cauchy problem for the Ostrovsky equation. Nonlinear Anal. TMA. 67, 1482–1503 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Isaza, P., Mejía, J.: Local well-posedness and quantitative ill-posedness for the Ostrovsky equation. Nonlinear Anal. TMA. 70, 2306–2316 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Isaza, P., Mejía, J.: On the support of solutions to the Ostrovsky equation with negative dispersion. J. Differ. Equ. 247, 1851–1865 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Isaza, P., Mejía, J.: On the support of solutions to the Ostrovsky equation with positive dispersion. Nonlinear Anal. TMA. 72, 4016–4029 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Isaza, P.: Unique continuation principle for the Ostrovsky equation with negative dispersion. J. Differ. Equ. 225, 796–811 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Kappeler, T., Topalov, P.: Global wellposedness of KdV in \(H^{-1}(T,R)\). Duke Math. J. 135, 327–360 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Kenig, C.E., Ponce, G., Vega, L.: Well-posedness of the initial value problem for the Korteweg-de Vries equation. J. Am. Math. Soc. 4, 323–347 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Kenig, C.E., Ponce, G., Vega, L.: The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices. Duke Math. J. 71, 1–21 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Kenig, C.E., Ponce, G., Vega, L.: Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Commun. Pure Appl. Math. 46, 527–620 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Kenig, C.E., Ponce, G., Vega, L.: A bilinear estimate with applications to the KdV equation. J. Am. Math. Soc. 9, 573–603 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Kenig, C.E., Ponce, G., Vega, L.: On the ill-posedness of some canonical dispersive equations. Duke Math. J. 106, 617–633 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Kishimoto, N.: Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity. Differ. Int. Equ. 22, 447–464 (2009)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Klainerman, S., Machedon, M.: Smoothing estimates for null forms and applications. Int. Math. Res. Notices 9, 383 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Leonov, A.: The effect of the earth’s rotation on the propagation of weak nonlinear surface and internal long oceanic waves. Ann. New York Acad. Sci. 373, 150–159 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Levandosky, S., Liu, Y.: Stability and weak rotation limit of solitary waves of the Ostrovsky equation. Discrete Contin. Dyn. Syst. Ser. B 7, 793–806 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Levandosky, S., Liu, Y.: Stability of solitary waves of a generalized Ostrovsky equation. SIAM J. Math. Anal. 38, 985–1011 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Levandosky, S., Liu, Y.: Stability and weak rotation limit of solitary waves of the Ostrovsky equation. Discret Contin. Dyn. Syst.-Ser. B 7, 793–806 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Linares, F., Milanés, A.: Local and global well-posedness for the Ostrovsky equation. J. Differ. Equ. 222, 325–340 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Li, Y.S., Huang, J.H., Yan, W.: The Cauchy problem for the Ostrovsky equation with negative dispersion at the critical regularity. J. Differ. Equ. 259, 1379–1408 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Li, Y.S., Wu, Y.F.: Global well-posedness for the Benjamin–Ono equation in low regularity. Nonlinear Anal. 73, 1610–1625 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Liu, B.P.: A priori bounds for KdV equation below \(H^{-3/4}\). J. Funct. Anal. 268, 501–554 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Liu, Y., Varlamov, V.: Stability of solitary waves and weak rotation limit for the Ostrovsky equation. J. Differ. Equ. 203, 159–183 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Liu, Y.: On the stability of solitary waves for the Ostrovsky equation. Q. Appl. Math. 65, 571–589 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Liu, Y., Pelinovsky, D., Sakovich, A.: Wave breaking in the Ostrovsky–Hunter equation. SIAM J. Math. Anal. 42, 1967–1985 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Molinet, L.: Sharp ill-posedness results for the KdV and mKdV equations on the torus. Adv. Math. 230, 1895–1930 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Molinet, L.: A note on ill posedness for the KdV equation. Differ. Int. Equ. 24, 759–765 (2011)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Muramatu, T., Taoka, S.: The initial value problem for the 1-D semilinear Schrödinger equation in Besov space. J. Math. Soc. Jpn 56, 853–888 (2004)CrossRefzbMATHGoogle Scholar
  56. 56.
    Rauch, J., Reed, M.: Nonlinear microlocal analysis of semilinear hyperbolic systems in one space dimension. Duke Math. J. 49, 397–475 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Shrira, V.: Propagation of long nonlinear waves in a layer of a rotating fluid, Iza. Akad. Sci. USSR Atmospher. Ocean. Phys. 17, 76–81 (1981)Google Scholar
  58. 58.
    Shrira, V.: On long essentially non-linear waves in a rotating ocean, Iza. Akad. Sci. USSR Atmospher. Ocean. Phys. 22, 395–405 (1986)Google Scholar
  59. 59.
    Tao, T.: Multilinear weighted convolution of \(L^2\) functions, and applications to non-linear dispersive equations. Am. J. Math. 123, 839–908 (2001)CrossRefzbMATHGoogle Scholar
  60. 60.
    Tsugawa, K.: Well-posedness and weak rotation limit for the Ostrovsky equation. J. Differ. Equ. 247, 3163–3180 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Tzvetkov, N.: Remark on the local ill-posedness for KdV equation. C. R. Acad. Sci. Paris 329, 1043–1047 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Ostrovskii, L.A.: Nonlinear internal waves in a rotating ocean. Okeanologiya 18, 181–191 (1978)Google Scholar
  63. 63.
    Varlamov, V., Liu, Y.: Cauchy problem for the Ostrovsky equation. Discret Cont. Dyn. Sys. A 10, 731–753 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Varlamov, V., Liu, Y.: Solitary waves and fundamental solution for Ostrovsky equation. Math. Comput. Simul. 69, 567–579 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Wang, H., Cui, S.B.: Well-posedness of the Cauchy problem of Ostrovsky equation in anisotropic Sobolev spaces. J. Math. Anal. Appl. 327(1), 88–100 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Zaiter, I.: Remarks on the Ostrovsky equation. Differ. Int. Equ. 20, 815–840 (2007)MathSciNetzbMATHGoogle Scholar
  67. 67.
    Zhang, P.Z., Liu, Y.: Symmetry and uniqueness of the solitary-wave solution for the Ostrovsky equation. Arch. Ration. Mech. Anal. 196, 811–837 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Wei Yan
    • 1
  • Yongsheng Li
    • 2
  • Jianhua Huang
    • 3
  • Jinqiao Duan
    • 4
  1. 1.College of Mathematics and Information ScienceHenan Normal UniversityXinxiangChina
  2. 2.School of MathematicsSouth China University of TechnologyGuangzhouChina
  3. 3.College of ScienceNational University of Defense TechnologyChangshaChina
  4. 4.Department of Applied MathematicsIllinois Institute of TechnologyChicagoUSA

Personalised recommendations