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

Guan, Chunxia; Yin, Zhaoyang

doi:10.1090/S0033-569X-2012-01267-8

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: 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