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Global weak solutions for a periodic two-component μ-Hunter–Saxton system

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Abstract

This paper is concerned with global existence of weak solutions for a periodic two-component μ-Hunter–Saxton system. We first derive global existence for strong solutions to the system with smooth approximate initial data. Then, we show that the limit of approximate solutions is a global weak solution of the two-component μ-Hunter–Saxton system.

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References

  1. Beals R., Sattinger D., Szmigielski J.: Inverse scattering solutions of the Hunter–Saxton equations. Appl. Anal. 78, 255–269 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bressan A., Constantin A.: Global solutions of the Hunter–Saxton equation. SIAM J. Math. Anal. 37, 996–1026 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Camassa R., Holm D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  4. Constantin A.: On the blow-up of solutions of a periodic shallow water equation. J. Nonlinear Sci. 10, 391–399 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Constantin A., Ivanov R.I.: On an integrable two-component Camass–Holm shallow water system. Phys. Lett. A. 372, 7129–7132 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Constantin A., Kolev B.: On the geometric approach to the motion of inertial mechanical systems. J. Phys. A. 35, R51–R79 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Constantin A., Lannes D.: The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations. Arch. Ration. Mech. Anal. 192, 165–186 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Constantin A., McKean H.P.: A shallow water equation on the circle. Commun. Pure Appl. Math. 52, 949–982 (1999)

    Article  MathSciNet  Google Scholar 

  9. Dai H.H., Pavlov M.: Transformations for the Camassa–Holm equation, its high-frequency limit and the Sinh–Gordon equation. J. Phys. Soc. Jpn. 67, 3655–3657 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. DiPerna R.J., Lions P.L.: Ordinary differential equations, transport theory and Sobolev space. Invent. Math. 98, 511–547 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  11. Escher, J.: Non-metric two-component Euler equations on the circle. Monatsh. Math. doi:10.1007/s00605-011-0323-3

  12. Escher, J., Kohlmann, M., Kolev, B.: Geometric aspects of the periodic μDP equation. http://arxiv.org/abs/1004.0978v1 (2010)

  13. Evans L.: Weak convergence methods for nonlinear partial differential equations. The American Mathematical Society Press, Rhode Island (1990)

    MATH  Google Scholar 

  14. Fokas, A., Fuchssteiner, B.: Symplectic structures, their Bäcklund transformations and hereditary symmetries. Phys. D 4, 47–66 (1981/1982)

    Google Scholar 

  15. Fu, Y., Liu, Y., Qu, C.: On the blow up structure for the generalized periodic Camassa–Holm and Degasperis–Procesi equations. http://arxiv.org/abs/1009.2466v2 (2010)

  16. Guan, C., Yin, Z.: Global weak solutions for a periodic two-component Hunter–Saxton system, Preprint

  17. Hunter J.K., Saxton R.: Dynamics of director fields. SIAM J. Appl. Math. 51, 1498–1521 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hunter J.K., Zheng Y.: On a completely integrable nonlinear hyperbolic variational equation. Phys. D. 79, 361–386 (1994)

    MathSciNet  MATH  Google Scholar 

  19. Hunter J., Zheng Y.: On a nonlinear hyperbolic variational equation: I, Global existence of weak solutions. Arch. Rat. Mech. Anal. 129, 305–353 (1995)

    Article  MathSciNet  Google Scholar 

  20. Hunter J., Zheng Y.: On a nonlinear hyperbolic variational equation: II, The zero-viscosity and dispersion limits. Arch. Rat. Mech. Anal. 129, 355–383 (1995)

    Article  MathSciNet  Google Scholar 

  21. Ivanov R.: Two component integrable systems modelling shallow water waves: the constant vorticity case. Wave Motion 46, 389–396 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Johnson R.S.: Camassa–Holm Korteweg-de Vries and related models for water waves. J. Fluid. Mech. 455, 63–82 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Khesin B., Lenells J., Misiolek G.: Generalized Hunter–Saxton equation and the geometry of the group of circle diffeomorphisms. Math. Ann. 342, 617–656 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lenells J., Misiolek G., Tiğlay F.: Integrable evolution equations on spaces of tensor densities and their peakon solutions. Commun. Math. Phys. 99, 129–161 (2010)

    Article  Google Scholar 

  25. Lions P.L.: Mathematical Topics in Fluid Mechanics, Vol. I. Incompressible Models. Oxford Lecture Series in Mathematics and Applications, 3. Clarendon. Oxford University Press, New York (1996)

    Google Scholar 

  26. Liu, J., Yin, Z.: Blow-up phenomena and global existence for a periodic two-component Hunter–Saxton system. http://arxiv.org/abs/1012.5448 (2010)

  27. Liu, J., Yin, Z.: On the Cauchy problem of a periodic 2-component μ-Hunter–Saxton system. Nonlinear Anal. doi:10.1016/j.na.2011.08.012

  28. Olver P., Rosenau P.: Tri-Hamiltonian duality between solitons and solitary wave solutions having compact support. Phys. Rev. E (3) 53, 1900–1906 (1996)

    Article  MathSciNet  Google Scholar 

  29. Simon J.: Compact sets in the space L P(0,T; B). Ann. Mat. Pura Appl. 146(4), 65–96 (1987)

    MathSciNet  MATH  Google Scholar 

  30. Wunsch M.: On the Hunter–Saxton system. Discret. Contin. Dyn. Syst. B. 12, 647–656 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wunsch M.: The generalized Hunter–Saxton system. SIAM J. Math. Anal. 42, 1286–1304 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Yin Z.: On the structure of solutions to the periodic Hunter–Saxton equation. SIAM J. Math. Anal. 36, 272–283 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhang P., Zheng Y.: On oscillations of an asymptotic equation of a nonlinear variational wave equation. Asymptot. Anal. 18, 307–327 (1998)

    MathSciNet  MATH  Google Scholar 

  34. Zhang P., Zheng Y.: On the existence and uniqueness of solutions to an asymptotic equation of a nonlinear variational wave equation. Acta Math. Sin. 15, 115–130 (1999)

    Article  MATH  Google Scholar 

  35. Zhang P., Zheng Y.: Existence and uniqueness of solutions to an asymptotic equation from a variational wave equation with general data. Arch. Rat. Mech. Anal. 155, 49–83 (2000)

    Article  MATH  Google Scholar 

  36. Zuo D.: A 2-component μ-Hunter–Saxton Equation. Inverse Probl. 26, 085003 (2010)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China

    Jingjing Liu & Zhaoyang Yin

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  1. Jingjing Liu
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  2. Zhaoyang Yin
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Correspondence to Jingjing Liu.

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Communicated by A. Constantin.

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Liu, J., Yin, Z. Global weak solutions for a periodic two-component μ-Hunter–Saxton system. Monatsh Math 168, 503–521 (2012). https://doi.org/10.1007/s00605-011-0346-9

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  • DOI: https://doi.org/10.1007/s00605-011-0346-9

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