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
Full-text PDF Free Access
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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.
References
- Richard Beals, David H. Sattinger, and Jacek Szmigielski, Inverse scattering solutions of the Hunter-Saxton equation, Appl. Anal. 78 (2001), no. 3-4, 255–269. MR 1883537, DOI https://doi.org/10.1080/00036810108840938
- Alberto Bressan and Adrian Constantin, Global solutions of the Hunter-Saxton equation, SIAM J. Math. Anal. 37 (2005), no. 3, 996–1026. MR 2191785, DOI https://doi.org/10.1137/050623036
- Roberto Camassa and Darryl D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993), no. 11, 1661–1664. MR 1234453, DOI https://doi.org/10.1103/PhysRevLett.71.1661
- Adrian Constantin and Rossen I. Ivanov, On an integrable two-component Camassa-Holm shallow water system, Phys. Lett. A 372 (2008), no. 48, 7129–7132. MR 2474608, DOI https://doi.org/10.1016/j.physleta.2008.10.050
- Adrian Constantin and Boris Kolev, On the geometric approach to the motion of inertial mechanical systems, J. Phys. A 35 (2002), no. 32, R51–R79. MR 1930889, DOI https://doi.org/10.1088/0305-4470/35/32/201
- Adrian Constantin and David Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal. 192 (2009), no. 1, 165–186. MR 2481064, DOI https://doi.org/10.1007/s00205-008-0128-2
- A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math. 52 (1999), no. 8, 949–982. MR 1686969, DOI https://doi.org/10.1002/%28SICI%291097-0312%28199908%2952%3A8%3C949%3A%3AAID-CPA3%3E3.0.CO%3B2-D
- Constantine M. Dafermos, Continuous solutions for balance laws, Ric. Mat. 55 (2006), no. 1, 79–91. MR 2248164, DOI https://doi.org/10.1007/s11587-006-0006-x
- R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989), no. 3, 511–547. MR 1022305, DOI https://doi.org/10.1007/BF01393835
- Hui-Hui Dai and Maxim Pavlov, Transformations for the Camassa-Holm equation, its high-frequency limit and the Sinh-Gordon equation, J. Phys. Soc. Japan 67 (1998), no. 11, 3655–3657. MR 1677529, DOI https://doi.org/10.1143/JPSJ.67.3655
- Joachim Escher, Olaf Lechtenfeld, and Zhaoyang Yin, Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation, Discrete Contin. Dyn. Syst. 19 (2007), no. 3, 493–513. MR 2335761, DOI https://doi.org/10.3934/dcds.2007.19.493
- Lawrence C. Evans, Weak convergence methods for nonlinear partial differential equations, CBMS Regional Conference Series in Mathematics, vol. 74, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990. MR 1034481
- B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D 4 (1981/82), no. 1, 47–66. MR 636470, DOI https://doi.org/10.1016/0167-2789%2881%2990004-X
- R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech. 455 (2002), 63–82. MR 1894796, DOI https://doi.org/10.1017/S0022112001007224
- John K. Hunter and Ralph Saxton, Dynamics of director fields, SIAM J. Appl. Math. 51 (1991), no. 6, 1498–1521. MR 1135995, DOI https://doi.org/10.1137/0151075
- John K. Hunter and Yu Xi Zheng, On a completely integrable nonlinear hyperbolic variational equation, Phys. D 79 (1994), no. 2-4, 361–386. MR 1306466, DOI https://doi.org/10.1016/0167-2789%2894%2990093-0
- John K. Hunter and Yu Xi Zheng, On a nonlinear hyperbolic variational equation. I. Global existence of weak solutions, Arch. Rational Mech. Anal. 129 (1995), no. 4, 305–353. MR 1361013, DOI https://doi.org/10.1007/BF00379259
- John K. Hunter and Yu Xi Zheng, On a nonlinear hyperbolic variational equation. II. The zero-viscosity and dispersion limits, Arch. Rational Mech. Anal. 129 (1995), no. 4, 355–383. MR 1361014, DOI https://doi.org/10.1007/BF00379260
- Jonatan Lenells, The Hunter-Saxton equation describes the geodesic flow on a sphere, J. Geom. Phys. 57 (2007), no. 10, 2049–2064. MR 2348278, DOI https://doi.org/10.1016/j.geomphys.2007.05.003
- Jonatan Lenells, The Hunter-Saxton equation: a geometric approach, SIAM J. Math. Anal. 40 (2008), no. 1, 266–277. MR 2403320, DOI https://doi.org/10.1137/050647451
- Pierre-Louis Lions, Mathematical topics in fluid mechanics. Vol. 1, Oxford Lecture Series in Mathematics and its Applications, vol. 3, The Clarendon Press, Oxford University Press, New York, 1996. Incompressible models; Oxford Science Publications. MR 1422251
- J. Liu and Z. Yin, Blow-up phenomena and global existence for a periodic Hunter-Saxton system, Preprint.
- Maxim V. Pavlov, The Gurevich-Zybin system, J. Phys. A 38 (2005), no. 17, 3823–3840. MR 2145381, DOI https://doi.org/10.1088/0305-4470/38/17/008
- Peter J. Olver and Philip Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E (3) 53 (1996), no. 2, 1900–1906. MR 1401317, DOI https://doi.org/10.1103/PhysRevE.53.1900
- Marcus Wunsch, On the Hunter-Saxton system, Discrete Contin. Dyn. Syst. Ser. B 12 (2009), no. 3, 647–656. MR 2525162, DOI https://doi.org/10.3934/dcdsb.2009.12.647
- Zhaoyang Yin, On the structure of solutions to the periodic Hunter-Saxton equation, SIAM J. Math. Anal. 36 (2004), no. 1, 272–283. MR 2083862, DOI https://doi.org/10.1137/S0036141003425672
- Ping Zhang and Yuxi Zheng, On oscillations of an asymptotic equation of a nonlinear variational wave equation, Asymptot. Anal. 18 (1998), no. 3-4, 307–327. MR 1668954
- Ping Zhang and Yuxi Zheng, On the existence and uniqueness of solutions to an asymptotic equation of a variational wave equation, Acta Math. Sin. (Engl. Ser.) 15 (1999), no. 1, 115–130. MR 1701136, DOI https://doi.org/10.1007/s10114-999-0063-7
- Ping Zhang and Yuxi Zheng, Existence and uniqueness of solutions of an asymptotic equation arising from a variational wave equation with general data, Arch. Ration. Mech. Anal. 155 (2000), no. 1, 49–83. MR 1799274, DOI https://doi.org/10.1007/s205-000-8002-2
References
- 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)
- A. Bressan and A. Constantin, Global solutions of the Hunter-Saxton equation, SIAM J. Math. Anal., 37 (2005), 996–1026. MR 2191785 (2006j:35203)
- R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661–1664. MR 1234453 (94f:35121)
- 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)
- 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)
- 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)
- A. Constantin and H. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949–982. MR 1686969 (2000m:37146)
- C. M. Dafermos, Continuous solutions for balance laws, Ricerche di Matematica, 55 (2006), 79–91. MR 2248164 (2007h:35222)
- 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)
- 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
- 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)
- 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)
- A. Fokas, and B. Fuchssteiner, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981/1982), 47–66. MR 636470 (84j:58046)
- R. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid. Mech., 457 (2002), 63–82. MR 1894796 (2003b:76026)
- J. Hunter and R. Saxton, Dynamics of director fields, SIAM J. Appl. Math., 51 (1991), 1498–1521. MR 1135995 (93a:76005)
- J. Hunter and Y. Zheng, On a completely integrable nonlinear hyperbolic variational equation, Phys. D, 79 (1994), 361–386. MR 1306466 (96e:35166)
- 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)
- 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)
- J. Lenells, The Hunter-Saxton equation describes the geodesic flow on a sphere, J. Geom. Phys., 57 (2007), 2049–2064. MR 2348278 (2008f:37169)
- J. Lenells, The Hunter-Saxton equation: a geometric approach, SIAM J. Math. Anal., 40 (2008), 266–277. MR 2403320 (2009b:37133)
- 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)
- J. Liu and Z. Yin, Blow-up phenomena and global existence for a periodic Hunter-Saxton system, Preprint.
- M. V. Pavlov, The Gurevich-Zybin system, J. Phys. A: Math. Gen., 38 (2005), 3823–3840. MR 2145381 (2005m:35241)
- 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)
- M. Wunsch, On the Hunter-Saxton system, Discret Contin. Dyn. Syst. Ser. B, 12 (2009), 647–656. MR 2525162 (2011a:35071)
- 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)
- 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)
- 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)
- 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
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.