Journal of Mathematical Fluid Mechanics

, Volume 14, Issue 3, pp 455–469 | Cite as

Global Existence for the Generalized Two-Component Hunter–Saxton System

Article

Abstract

We study the global existence of solutions to a two-component generalized Hunter–Saxton system in the periodic setting. We first prove a persistence result for the solutions. Then for some particular choices of the parameters (α, κ), we show the precise blow-up scenarios and the existence of global solutions to the generalized Hunter–Saxton system under proper assumptions on the initial data. This significantly improves recent results.

Mathematics Subject Classification (2010)

Primary 35B10 Secondary 35Q35 

Keywords

Generalized Hunter–Saxton system Global existence Blow-up scenario 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alvarez-Samaniego B., Lannes D.: Large time existence for 3D water-waves and asymptotics. Invent. Math. 171(3), 485–541 (2008)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Beals R., Sattinger D.H., Szmigielski J.: Inverse scattering solutions of the Hunter–Saxton equation. Appl. Anal. 78(3&4), 255–269 (2001)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bressan A., Constantin A.: Global solutions of the Hunter–Saxton equation. SIAM J. Math. Anal. 37(3), 996–1026 (2005)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bressan A., Holden H., Raynaud X.: Lipschitz metric for the Hunter–Saxton equation. J. Math. Pure Appl. (9) 94(1), 68–92 (2010)MathSciNetMATHGoogle Scholar
  5. 5.
    Chae D.: On the blow-up problem for the axisymmetric 3D Euler equations. Nonlinearity 21, 2053–2060 (2008)MathSciNetADSMATHCrossRefGoogle Scholar
  6. 6.
    Chae, D., Okamoto, H.: Nonstationary von Kármán–Batchelor flow. Preprint (2009)Google Scholar
  7. 7.
    Cho C.-H., Wunsch M.: Global and singular solutions to the generalized Proudman–Johnson equation. J. Diff. Equ. 249(2), 392–413 (2010)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Constantin A.: The trajectories of particles in Stokes waves. Invent. Math. 166, 523–535 (2006)MathSciNetADSMATHCrossRefGoogle Scholar
  9. 9.
    Constantin A., Escher J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181, 229–243 (1998)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Constantin A., Ivanov R.I.: On an integrable two-component Camassa–Holm shallow water system. Phys. Lett. A 372, 7129–7132 (2008)MathSciNetADSMATHCrossRefGoogle Scholar
  11. 11.
    Constantin A., Kolev B.: Geodesic flow on the diffeomorphism group of the circle. Comment. Math. Helv. 78, 787–804 (2003)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Constantin A., Kolev B.: Integrability of invariant metrics on the diffeomorphism group of the circle. J. Nonlinear Sci. 16, 109–122 (2006)MathSciNetADSMATHCrossRefGoogle Scholar
  13. 13.
    Constantin A., Kolev B.: On the geometric approach to the motion of inertial mechanical systems. J. Phys. A Math. Gen. 35, R51–R79 (2002)MathSciNetADSMATHCrossRefGoogle Scholar
  14. 14.
    Constantin A., Lannes D.: The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations. Arch. Ration. Mech. Anal. 192(1), 165–186 (2009)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Constantin A., Wunsch M.: On the inviscid Proudman–Johnson equation. Proc. Jpn. Acad. Ser. A Math. Sci. 85(7), 81–83 (2009)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Constantin P., Lax P.D., Majda A.: A simple one-dimensional model for the three-dimensional vorticity equation. Comm. Pure Appl. Math. 38, 715–724 (1985)MathSciNetADSMATHCrossRefGoogle Scholar
  17. 17.
    Escher J., Kohlmann M., Lenells J.: The geometry of the two-component Camassa–Holm and Degasperis–Procesi equations. J. Geom. Phys. 61(2), 436–452 (2011)MathSciNetADSMATHCrossRefGoogle Scholar
  18. 18.
    Escher J., Lechtenfeld O., Yin Z.: Well-posedness and blow up phenomena for the 2-component Camassa–Holm equations. Discr. Contin. Dyn. Syst. 19(3), 493–513 (2007)MathSciNetMATHGoogle Scholar
  19. 19.
    Guan C., Yin Z.: Global existence and blow-up phenomena for an integrable two-component Camassa–Holm shallow water system. J. Diff. Equ. 248(8), 2003–2014 (2010)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Guo Z.: Blow up and global solutions to a new integrable model with two components. J. Math. Anal. Appl. 372(1), 316–327 (2010)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Guo Z., Zhou Y.: On Solutions to a two-component generalized Camassa–Holm equation. Stud. Appl. Math. 124(3), 307–322 (2010)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Holm D.D., Staley M.F.: Wave structure and nonlinear balances in a family of evolutionary PDEs. SIAM J. Appl. Dyn. Syst. 2(3), 323–380 (2003)MathSciNetADSMATHCrossRefGoogle Scholar
  23. 23.
    Hou T.Y., Li C.: Dynamic stability of the three-dimensional axisymmetric Navier–Stokes equations with swirl. Comm. Pure Appl. Math. LXI, 661–697 (2008)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Hunter J.K., Saxton R.: Dynamics of director fields. SIAM J. Appl. Math. 51, 1498–1521 (1991)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Johnson R.S.: Camassa–Holm, Korteweg–de Vries and related models for water waves. J. Fluid Mech. 455, 63–82 (2002)MathSciNetADSMATHCrossRefGoogle Scholar
  26. 26.
    Khesin B., Misiołek G.: Euler equations on homogeneous spaces and Virasoro orbits. Adv. Math. 176(1), 116–144 (2003)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Kouranbaeva S.: The Camassa–Holm equation as a geodesic flow on the diffeomorphism group. J. Math. Phys. 40(2), 857–868 (1999)MathSciNetADSMATHCrossRefGoogle Scholar
  28. 28.
    Lenells J.: The Hunter–Saxton equation describes the geodesic flow on a sphere. J. Geom. Phys. 57, 2049–2064 (2007)MathSciNetADSMATHCrossRefGoogle Scholar
  29. 29.
    Lenells J.: The Hunter–Saxton equation: a geometric approach. SIAM J. Math. Anal. 40, 266–277 (2008)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Lenells J.: Weak geodesic flow and global solutions of the Hunter–Saxton equation. Discr. Contin. Dyn. Syst. 18(4), 643–656 (2007)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Lenells J., Lechtenfeld O.: On the N = 2 supersymmetric Camassa–Holm and Hunter–Saxton systems. J. Math. Phys. 50, 1–17 (2009)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Misiołek G.: A shallow water equation as a geodesic flow on the Bott–Virasoro group and the KdV equation. Proc. Am. Math. Soc. 125, 203–208 (1998)Google Scholar
  33. 33.
    Misiołek G.: Classical solutions of the periodic Camassa–Holm equation. Geom. Funct. Anal. 12(5), 1080–1104 (2002)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Mohajer K.: A note on traveling wave solutions to the two-component Camassa–Holm equation. J. Nonlinear Math. Phys. 16, 117–125 (2009)MathSciNetADSMATHCrossRefGoogle Scholar
  35. 35.
    Mustafa O.G.: On smooth traveling waves of an integrable two-component Camassa–Holm equation. Wave Motion 46, 397–402 (2009)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Okamoto H.: Well-posedness of the generalized Proudman–Johnson equation without viscosity. J. Math. Fluid Mech. 11, 46–59 (2009)MathSciNetADSMATHCrossRefGoogle Scholar
  37. 37.
    Okamoto H., Ohkitani K.: On the role of the convection term in the equations of motion of incompressible fluid. J. Phys. Soc. Jpn. 74, 2737–2742 (2005)ADSMATHCrossRefGoogle Scholar
  38. 38.
    Okamoto H., Sakajo T., Wunsch M.: On a generalization of the Constantin–Lax–Majda equation. Nonlinearity 21, 2447–2461 (2008)MathSciNetADSMATHCrossRefGoogle Scholar
  39. 39.
    Okamoto, H., Zhu, J.: Some similarity solutions of the Navier–Stokes equations and related topics. In: Proceedings of 1999 International Conference on Nonlinear Analysis (Taipei), Taiwanese J. Math. vol. 4, pp. 65–103 (2000)Google Scholar
  40. 40.
    Pavlov M.V.: The Gurevich–Zybin system. J. Phys. A Math. Gen. 38, 3823–3840 (2005)ADSCrossRefGoogle Scholar
  41. 41.
    Proudman I., Johnson K.: Boundary-layer growth near a rear stagnation point. J. Fluid Mech. 12, 161–168 (1962)MathSciNetADSMATHCrossRefGoogle Scholar
  42. 42.
    Saxton R., Tıuglay F.: Global existence of some infinite energy solutions for a perfect incompressible fluid. SIAM J. Math. Anal. 4, 1499–1515 (2008)CrossRefGoogle Scholar
  43. 43.
    Taylor M.: Pseudodifferential operators and nonlinear PDE. Progress in Mathematics, vol. 100. Birkhäuser Boston, Inc., Boston (1991)CrossRefGoogle Scholar
  44. 44.
    Tıuglay F.: The periodic Cauchy problem of the modified Hunter–Saxton equation. J. Evol. Equ. 5(4), 509–527 (2005)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Toland J.F.: Stokes waves. Topol. Methods Nonlinear Anal. 7, 1–48 (1996)MathSciNetMATHGoogle Scholar
  46. 46.
    Whitham G.B.: Linear and nonlinear waves. Wiley, New York (1999)MATHCrossRefGoogle Scholar
  47. 47.
    Wunsch M.: On the Hunter–Saxton system. Discr. Contin. Dyn. Syst. B 12(3), 647–656 (2009)MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    Wunsch M.: The generalized Hunter–Saxton system. SIAM J. Math. Anal. 42(3), 1286–1304 (2010)MathSciNetMATHCrossRefGoogle Scholar
  49. 49.
    Wunsch M.: The generalized Proudman–Johnson equation revisited. J. Math. Fluid Mech. 13(1), 147–154 (2011)MathSciNetADSCrossRefGoogle Scholar
  50. 50.
    Wunsch M.: Weak geodesic flow on a semi-direct product and global solutions to the periodic Hunter–Saxton system. Nonlinear Anal. 74, 4951–4960 (2011)MathSciNetMATHCrossRefGoogle Scholar
  51. 51.
    Yin Z.: On the structure of solutions to the periodic Hunter–Saxton equation. SIAM J. Math. Anal. 36(1), 272–283 (2004)MathSciNetMATHCrossRefGoogle Scholar
  52. 52.
    Zhang P., Liu Y.: Stability of solitary waves and wave-breaking phenomena for the two-component Camassa–Holm system. Int. Math. Res. Notices 11, 1981–2021 (2010)Google Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical SciencesFudan UniversityShanghaiPeople’s Republic of China
  2. 2.Department of MathematicsSwiss Federal Institute of Technology ZurichZurichSwitzerland

Personalised recommendations