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Well-posedness for a modified two-component Camassa-Holm system in critical spaces
Global conservative and dissipative solutions of the generalized Camassa-Holm equation
1. | College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China, China |
References:
[1] |
R. Beals, D. Sattinger and J. Szmigielski, Acoustic scattering and the extended Korteweg-de Vries hierarchy, Adv. Math., 140 (1998), 190-206. |
[2] |
A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Rational Mech. Anal., 183 (2007), 215-239. |
[3] |
A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl. (Singap.), 5 (2007), 1-27.
doi: 10.1142/S0219530507000857. |
[4] |
A. Boutet de Monvel and D. Shepelsky, Riemann-Hilbert approach for the Camassa-Holm equation on the line, C. R. Math. Acad. Sci. Paris, 343 (2006), 627-632. |
[5] |
A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. Roy. Soc. London(A), 457 (2001), 953-970. |
[6] |
A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. |
[7] |
R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. |
[8] |
R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33.
doi: 10.1016/S0065-2156(08)70254-0. |
[9] |
G. M. Coclite, H. Holden and K. H. Karlsen, Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM J. Math. Anal., 37 (2005), 1044-1069. |
[10] |
A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568. |
[11] |
A. Constantin, V. Gerdjikov and R. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207. |
[12] |
A. Constantin and W. A. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610. |
[13] |
A. Constantin and W. A. Strauss, Stability of a class of solitary waves in compressible elastic rods, Phys. Lett. A, 270 (2000), 140-148.
doi: 10.1016/S0375-9601(00)00255-3. |
[14] |
A. Constantin and W. A. Strauss, Stability of the Camassa-Holm solitons, J. Nonlinear. Sci., 12 (2002), 415-422. |
[15] |
H. H. Dai and Y. Huo, Solitary shock waves and other travelling waves in a general compressible hyperelastic rod, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 456 (2000), 331-363.
doi: 10.1098/rspa.2000.0520. |
[16] |
A. Fokas and B. Fuchssteiner, Symplectic structures, their Backlund transformations and hereditary symmetries, Phys. D, 4 (1981), 47-66. |
[17] |
H. Holden and X. Raynaud, Global conservative solutions of the Camassa-Holm equation-a Lagrangian point of view, Comm. Partial Differential Equations, 32 (2007) 1511-1549. |
[18] |
H. Holden and X. Raynaud, Global conservative solutions of the generalized hyperelastic-rod wave equation, J. Differential Equations, 233 (2007), 448-484. |
[19] |
H. Holden and X. Raynaud, Dissipative solutions for the Camassa-Holm equation, Discrete Contin. Dyn. Syst., 24 (2009), 1047-1112. |
[20] |
J. Lenells, Conservation laws of the Camassa-Holm equation, J. Phys. A, 38 (2005), 869-880.
doi: 10.1088/0305-4470/38/4/007. |
[21] |
O. G. Mustafa, On the Cauchy problem for a generalized Camassa-Holm equation, Nonlinear Anal. TMA, 64 (2006), 1382-1399. |
[22] |
O. G. Mustafa, Solitary waves for a generalized Camassa-Holm equation, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 14 (2007), 205-212. |
[23] |
O. G. Mustafa, Global conservative solutions of the hyperelastic rod equation, Int. Math. Res. Notices, (2007), Art. ID rnm 040, 26 pp. |
[24] |
O. G. Mustafa, Global dissipative solution of the generalized Camassa-Holm equation, J. Nonlinear Math. Phys., 15 (2008), 96-115. |
[25] |
L. Tian and X. Song, New peaked solitary wave solutions of the generalized Camassa-Holm equation, Chaos Solitons Fractals, 19 (2004), 621-637. |
[26] |
J. Shen and W. Xu, Bifurcations of smooth and non-smooth travelling wave solutions in the generalized Camassa-Holm equation, Chaos Solitons Fractals, 26 (2005), 1149-1162. |
[27] |
Z. Yin, On the Cauchy problem for a nonlinearly dispersive wave equation, J. Nonlinear Math. Phys., 10 (2003), 10-15.
doi: 10.2991/jnmp.2003.10.1.2. |
[28] |
Z. Yin, On the Cauchy problem for the generalized Camassa-Holm equation, Nonlinear Anal. TMA, 66 (2007), 460-471. |
[29] |
Z. Yin, On the blow-up scenario for the generalized Camassa-Holm equation, Comm. Partial Differential Equations, 29 (2004), 867-877. |
show all references
References:
[1] |
R. Beals, D. Sattinger and J. Szmigielski, Acoustic scattering and the extended Korteweg-de Vries hierarchy, Adv. Math., 140 (1998), 190-206. |
[2] |
A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Rational Mech. Anal., 183 (2007), 215-239. |
[3] |
A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl. (Singap.), 5 (2007), 1-27.
doi: 10.1142/S0219530507000857. |
[4] |
A. Boutet de Monvel and D. Shepelsky, Riemann-Hilbert approach for the Camassa-Holm equation on the line, C. R. Math. Acad. Sci. Paris, 343 (2006), 627-632. |
[5] |
A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. Roy. Soc. London(A), 457 (2001), 953-970. |
[6] |
A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. |
[7] |
R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. |
[8] |
R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33.
doi: 10.1016/S0065-2156(08)70254-0. |
[9] |
G. M. Coclite, H. Holden and K. H. Karlsen, Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM J. Math. Anal., 37 (2005), 1044-1069. |
[10] |
A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568. |
[11] |
A. Constantin, V. Gerdjikov and R. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207. |
[12] |
A. Constantin and W. A. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610. |
[13] |
A. Constantin and W. A. Strauss, Stability of a class of solitary waves in compressible elastic rods, Phys. Lett. A, 270 (2000), 140-148.
doi: 10.1016/S0375-9601(00)00255-3. |
[14] |
A. Constantin and W. A. Strauss, Stability of the Camassa-Holm solitons, J. Nonlinear. Sci., 12 (2002), 415-422. |
[15] |
H. H. Dai and Y. Huo, Solitary shock waves and other travelling waves in a general compressible hyperelastic rod, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 456 (2000), 331-363.
doi: 10.1098/rspa.2000.0520. |
[16] |
A. Fokas and B. Fuchssteiner, Symplectic structures, their Backlund transformations and hereditary symmetries, Phys. D, 4 (1981), 47-66. |
[17] |
H. Holden and X. Raynaud, Global conservative solutions of the Camassa-Holm equation-a Lagrangian point of view, Comm. Partial Differential Equations, 32 (2007) 1511-1549. |
[18] |
H. Holden and X. Raynaud, Global conservative solutions of the generalized hyperelastic-rod wave equation, J. Differential Equations, 233 (2007), 448-484. |
[19] |
H. Holden and X. Raynaud, Dissipative solutions for the Camassa-Holm equation, Discrete Contin. Dyn. Syst., 24 (2009), 1047-1112. |
[20] |
J. Lenells, Conservation laws of the Camassa-Holm equation, J. Phys. A, 38 (2005), 869-880.
doi: 10.1088/0305-4470/38/4/007. |
[21] |
O. G. Mustafa, On the Cauchy problem for a generalized Camassa-Holm equation, Nonlinear Anal. TMA, 64 (2006), 1382-1399. |
[22] |
O. G. Mustafa, Solitary waves for a generalized Camassa-Holm equation, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 14 (2007), 205-212. |
[23] |
O. G. Mustafa, Global conservative solutions of the hyperelastic rod equation, Int. Math. Res. Notices, (2007), Art. ID rnm 040, 26 pp. |
[24] |
O. G. Mustafa, Global dissipative solution of the generalized Camassa-Holm equation, J. Nonlinear Math. Phys., 15 (2008), 96-115. |
[25] |
L. Tian and X. Song, New peaked solitary wave solutions of the generalized Camassa-Holm equation, Chaos Solitons Fractals, 19 (2004), 621-637. |
[26] |
J. Shen and W. Xu, Bifurcations of smooth and non-smooth travelling wave solutions in the generalized Camassa-Holm equation, Chaos Solitons Fractals, 26 (2005), 1149-1162. |
[27] |
Z. Yin, On the Cauchy problem for a nonlinearly dispersive wave equation, J. Nonlinear Math. Phys., 10 (2003), 10-15.
doi: 10.2991/jnmp.2003.10.1.2. |
[28] |
Z. Yin, On the Cauchy problem for the generalized Camassa-Holm equation, Nonlinear Anal. TMA, 66 (2007), 460-471. |
[29] |
Z. Yin, On the blow-up scenario for the generalized Camassa-Holm equation, Comm. Partial Differential Equations, 29 (2004), 867-877. |
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