Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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


Authors: Chunxia Guan and Zhaoyang Yin
Journal: Quart. Appl. Math. 70 (2012), 285-297
MSC (2010): Primary 35G25, 35L05
DOI: https://doi.org/10.1090/S0033-569X-2012-01267-8
Published electronically: February 3, 2012
MathSciNet review: 2953104
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Abstract | References | Similar Articles | Additional Information

Abstract: This paper is concerned with global existence of weak solutions for a periodic two-component 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 Hunter-Saxton system.


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  • 1. R. Beals, D. Sattinger and J. Szmigielski, Inverse scattering solutions of the Hunter-Saxton equations, Appl. Anal., 78 (2001), 255-269. MR 1883537 (2002m:35187)
  • 2. A. Bressan and A. Constantin, Global solutions of the Hunter-Saxton equation, SIAM J. Math. Anal., 37 (2005), 996-1026. MR 2191785 (2006j:35203)
  • 3. R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. MR 1234453 (94f:35121)
  • 4. A. Constantin and R. I. Ivanov, On an integrable two-component Camass-Holm shallow water system, Phys. Lett. A, 372 (2008), 7129-7132. MR 2474608 (2009m:35418)
  • 5. A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems, J. Phys. A, 35 (2002), R51-R79. MR 1930889 (2003g:37138)
  • 6. A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Rat. Mech. Anal., 1992 (2009), 165-186. MR 2481064 (2010f:35334)
  • 7. A. Constantin and H. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982. MR 1686969 (2000m:37146)
  • 8. C. M. Dafermos, Continuous solutions for balance laws, Ricerche di Matematica, 55 (2006), 79-91. MR 2248164 (2007h:35222)
  • 9. R. J. DiPerna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev space, Invent. Math., 98 (1989), 511-547. MR 1022305 (90j:34004)
  • 10. H. H. Dai and M. Pavlov, Transformations for the Camassa-Holm equation, its high-frequency limit and the Sinh-Gordon equation, J. P. Soc. Japan, 67 (1998), 3655-3657. MR 1677529
  • 11. J. Escher, O. Lechtenfeld and Z. Yin, Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation, Discrete Contin. Dyn. Syst., 19 (2007), 493-513. MR 2335761 (2008j:35154)
  • 12. L. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS Regional Conference Series in Mathematics, vol. 74, American Mathematical Society, Rhode Island, 1990. MR 1034481 (91a:35009)
  • 13. A. Fokas, and B. Fuchssteiner, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981/1982), 47-66. MR 636470 (84j:58046)
  • 14. R. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid. Mech., 457 (2002), 63-82. MR 1894796 (2003b:76026)
  • 15. J. Hunter and R. Saxton, Dynamics of director fields, SIAM J. Appl. Math., 51 (1991), 1498-1521. MR 1135995 (93a:76005)
  • 16. J. Hunter and Y. Zheng, On a completely integrable nonlinear hyperbolic variational equation, Phys. D, 79 (1994), 361-386. MR 1306466 (96e:35166)
  • 17. J. Hunter and Y. Zheng, On a nonlinear hyperbolic variational equation: I, Global existence of weak solutions, Arch. Rat. Mech. Anal., 129 (1995), 305-353. MR 1361013 (96m:35215)
  • 18. J. Hunter and Y. Zheng, On a nonlinear hyperbolic variational equation: II, The zero-viscosity and dispersion limits, Arch. Rat. Mech. Anal., 129 (1995), 355-383. MR 1361014 (96m:35216)
  • 19. J. Lenells, The Hunter-Saxton equation describes the geodesic flow on a sphere, J. Geom. Phys., 57 (2007), 2049-2064. MR 2348278 (2008f:37169)
  • 20. J. Lenells, The Hunter-Saxton equation: a geometric approach, SIAM J. Math. Anal., 40 (2008), 266-277. MR 2403320 (2009b:37133)
  • 21. P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. I. Incompressible Models. Oxford Lecture Series in Mathematics and Applications, 3. Clarendon, Oxford University Press, New York, 1996. MR 1422251 (98b:76001)
  • 22. J. Liu and Z. Yin, Blow-up phenomena and global existence for a periodic Hunter-Saxton system, Preprint.
  • 23. M. V. Pavlov, The Gurevich-Zybin system, J. Phys. A: Math. Gen., 38 (2005), 3823-3840. MR 2145381 (2005m:35241)
  • 24. P. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary wave solutions having compact support, Phys. Rev. E (3), 53 (1996), 1900-1906. MR 1401317 (97c:35172)
  • 25. M. Wunsch, On the Hunter-Saxton system, Discret Contin. Dyn. Syst. Ser. B, 12 (2009), 647-656. MR 2525162 (2011a:35071)
  • 26. Z. Yin, On the structure of solutions to the periodic Hunter-Saxton equation, SIAM J. Math.Anal., 36 (2004), 272-283. MR 2083862 (2005e:35034)
  • 27. P. Zhang and Y. Zheng, On oscillations of an asymptotic equation of a nonlinear variational wave equation, Asymptot. Anal., 18 (1998), 307-327. MR 1668954 (99j:35146)
  • 28. P. Zhang and Y. Zheng, On the existence and uniqueness of solutions to an asymptotic equation of a nonlinear variational wave equation, Acta Math. Sinica, 15 (1999), 115-130. MR 1701136 (2000f:35090)
  • 29. P. Zhang and Y. Zheng, Existence and uniqueness of solutions to an asymptotic equation from a variational wave equation with general data, Arch. Rat. Mech. Anal., 155 (2000), 49-83. MR 1799274 (2001j:35184)

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Additional Information

Chunxia Guan
Affiliation: Institut Franco-Chinois de l’Energie Nucléaire
Email: guanchunxia123@163.com

Zhaoyang Yin
Affiliation: Department of Mathematics, Sun Yat-sen University, 510275 Guangzhou, China
Email: mcsyzy@mail.sysu.edu.cn

DOI: https://doi.org/10.1090/S0033-569X-2012-01267-8
Keywords: Periodic two-component Hunter-Saxton system, weak solutions, global existence, approximate solutions.
Received by editor(s): July 29, 2010
Published electronically: February 3, 2012
Article copyright: © Copyright 2012 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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