Global existence for the periodic dispersive Hunter–Saxton equation

  • Weikui Ye
  • Zhaoyang YinEmail author


In this paper, we study an integrable dispersive Hunter–Saxton equation in periodic domain. Firstly, we establish the local well-posedness of the Cauchy problem of the equation in \(H^s ({\mathbb {S}}), s \ge 2,\) by applying the Kato method. Then, based on a sign-preserve property, we obtain a global existence result for the equation. Moreover, we extend the obtained result to some periodic nonlinear partial differential equations of second order of the general form.


The periodic dispersive Hunter–Saxton equation Local well-posedness The Kato method Global existence 

Mathematics Subject Classification

35A01 35L03 35L05 35L60 



This work was partially supported by NNSFC (No. 11671407), FDCT (No. 0091/2018/A3), Guangdong Special Support Program (No. 8-2015) and the key Project of NSF of Guangdong Province (No. 2016A030311004). The authors thank the referee for valuable comments and suggestions.


  1. 1.
    Beals, R., Sattinger, D., Szmigielski, J.: Inverse scattering solutions of the Hunter–Saxton equations. Appl. Anal. 78, 255–269 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    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 near-corner waves). Eur. J. Appl. Math. 16, 65–81 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Boyd, J.P.: Microbreaking and polycnoidal waves in the Ostrovsky–Hunter equation. Phys. Lett. A 338, 36–43 (2005)CrossRefzbMATHGoogle Scholar
  4. 4.
    Bressan, A., Constantin, A.: Global solutions of the Hunter–Saxton equation. SIAM J. Math. Anal. 37, 996–1026 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bressan, A., Constantin, A.: Global conservative solutions of the Camassa–Holm equation. Arch. Ration. Mech. Anal. 183, 215–239 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bressan, A., Constantin, A.: Global dissipative solutions of the Camassa–Holm equation. Anal. Appl. 5, 1–27 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Constantin, A.: On the scattering problem for the Camassa–Holm equation. Proc. R. Soc. Lond. A 457, 953–970 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Constantin, A.: Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Ann. Inst. Fourier (Grenoble) 50, 321–362 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Constantin, A.: The trajectories of particles in Stokes waves. Invent. Math. 166, 523–535 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Constantin, A.: Particle trajectories in extreme Stokes waves. IMA J. Appl. Math. 77, 293–307 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Constantin, A.: The Hamiltonian structure of the Camassa–Holm equation. Expo. Math. 15(1), 53–85 (1997)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Constantin, A., Escher, J.: Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation. Commun. Pure Appl. Math. 51, 475–504 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Constantin, A., Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181, 229–243 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Constantin, A., Escher, J.: Particle trajectories in solitary water waves. Bull. Am. Math. Soc. 44, 423–431 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Constantin, A., Escher, J.: Analyticity of periodic traveling free surface water waves with vorticity. Ann. Math. 173, 559–568 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Constantin, A., Gerdjikov, V.S., Ivanov, R.I.: Inverse scattering transform for the Camassa–Holm equation. Inverse Probl. 22, 2197–2207 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Constantin, A., McKean, H.P.: A shallow water equation on the circle. Commun. Pure Appl. Math. 52, 949–982 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Constantin, A., Molinet, L.: Global weak solutions for a shallow water equation. Commun. Math. Phys. 211, 45–61 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Constantin, A., Strauss, W.A.: Stability of peakons. Commun. Pure Appl. Math. 53, 603–610 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Dai, H .H., Pavlov, M.: Transformations for the Camassa–Holm equation, its high-frequency limit and the Sinh–Gordon equation. J. P. Soc. Jpn. 67, 3655–3657 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Danchin, R.: A note on well-posedness for Camassa–Holm equation. J. Differ. Equ. 192, 429–444 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Fokas, A., Fuchssteiner, B.: Symplectic structures, their Bäcklund transformation and hereditary symmetries. Phys. D 4, 47–66 (1981/82)Google Scholar
  23. 23.
    Guo, Z., Liu, X., Molinet, L., Yin, Z.: Ill-posedness of the Camassa Holm and related equations in the critical space. J. Differ. Equ. 266, 1698–1707 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Grimshaw, R., Pelinovsky, D.: Global existence of small-norm solutions in the reduced Ostrovsky equation. Discret. Contin. Dyn. Syst. Ser. 34, 557–566 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Hone, A., Novikov, V., Wang, J.: Generalizations of the short pulse equation. Lett. Math. Phys. 108, 927–947 (2018)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Hunter, J.: Numerical solutions of some nonlinear dispersive wave equations. In: Computational Solution of Nonlinear Systems of Equations. Lectures in Applied Mathematics, vol. 26, pp. 301–316. AMS, Providence (1990)Google Scholar
  27. 27.
    Hunter, J.K., Saxton, R.: Dynamics of director fields. SIAM J. Appl. Math. 51, 1498–1521 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Hunter, J.K., Zheng, Y.: On a completely integrable nonlinear hyperbolic variational equation. Phys. D 79, 361–386 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Kato, T.: Quasi-linear equations of evolution, with applications to partial differential equations. In: Spectral Theory and Differential Equations. Lecture Notes in Mathematics, vol. 448, pp. 25–70. Springer, Berlin (1975)Google Scholar
  30. 30.
    Li, M., Yin, Z.: Blow-up phenomena and travelling wave solutions to the periodic integrable dispersive Hunter–Saxton equation. Discret. Contin. Dyn. Syst. Ser. 37, 6471–6485 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Li, J., Yin, Z.: Remarks on the well-posedness of Camassa–Holm type equations in Besov spaces. J. Differ. Equ. 261(11), 6125–6143 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Li, J.Y., Zhang, K.J.: Global existence of dissipative solutions to the Hunter–Saxton equation via vanishing viscosity. J. Differ. Equ. 250, 1427–1447 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Liu, Y., Pelinovsky, D., Sakovich, A.: Wave breaking in the Ostrovsky Hunter equation. SIAM J. Math. Anal. 42, 1967–1985 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Liu, Y., Pelinovsky, D., Sakovich, A.: Wave breaking in the short-pulse equation. Dyn. Partial Differ. Equ. 6, 291–310 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Morrison, A.J., Parkes, E.J., Vakhnenko, V.O.: The N loop soliton solutions of the Vakhnenko equation. Nonlinearity 12, 1427–1437 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Olver, P., Rosenau, P.: Tri-Hamiltonian duality between solitions and solitary wave solutions having compact support. Phys. Rev. E 53, 1900–1906 (1996)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Pelinovsky, D., Sakovich, A.: Global well-posedness of the short-pulse and sine-Gordon equations in energy space. Commun. Partial Differ. Equ. 35, 613–629 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Parkes, E.J.: Explicit solutions of the reduced Ostrovsky equation. Chaos Solitons Fractals 31, 181–191 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Rodríguez-Blanco, G.: On the Cauchy problem for the Camassa–Holm equation. Nonlinear Anal. 46(3), 309–327 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Schäfter, T., Wayne, C.E.: Propagation of ultra-short optical pulses in cubic nonlinear media. Phys. D 196, 90–105 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Stefanov, A., Shen, Y., Kevrekidis, P.G.: Well-posedness and small data scattering for the generalized Ostrovsky equation. J. Differ. Equ. 249, 2600–2617 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Toland, J.F.: Stokes waves. Topol. Methods Nonlinear Anal. 7, 1–48 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Xin, Z., Zhang, P.: On the weak solutions to a shallow water equation. Commun. Pure Appl. Math. 53, 1411–1433 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Yin, Z.: On the structure of solutions to the periodic Hunter–Saxton equation. SIAM J. Math. Anal. 36, 272–283 (2004)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsSun Yat-sen UniversityGuangzhouChina
  2. 2.Faculty of Information TechnologyMacau University of Science and TechnologyTaipaChina

Personalised recommendations