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Smoothness of the flow map for low-regularity solutions of the Camassa-Holm equations
Lipschitz metric for the Camassa--Holm equation on the line
1. | Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim |
2. | Centre of Mathematics for Applications, University of Oslo, NO-0316 Oslo |
References:
[1] |
A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation,, Arch. Ration. Mech. Anal., 183 (2007), 215.
doi: 10.1007/s00205-006-0010-z. |
[2] |
A. Bressan, H. Holden and X. Raynaud, Lipschitz metric for the Hunter-Saxton equation,, J. Math. Pures Appl., 94 (2010), 68.
doi: 10.1016/j.matpur.2010.02.005. |
[3] |
R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solutions,, Phys. Rev. Lett., 71 (1993), 1661.
doi: 10.1103/PhysRevLett.71.1661. |
[4] |
R. Camassa, D. D. Holm and J. Hyman, A new integrable shallow water equation,, Adv. Appl. Mech., 31 (1994), 1. Google Scholar |
[5] |
A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 303.
|
[6] |
A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181 (1998), 229.
doi: 10.1007/BF02392586. |
[7] |
A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation,, Math. Z., 233 (2000), 75.
doi: 10.1007/PL00004793. |
[8] |
A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Rat. Mech. Anal., 192 (2009), 165.
doi: 10.1007/s00205-008-0128-2. |
[9] |
H.-H. Dai, Exact traveling-wave solutions of an integrable equation arising in hyperelastic rods,, Wave Motion, 28 (1998), 367.
doi: 10.1016/S0165-2125(98)00014-6. |
[10] |
H.-H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod,, Acta Mech., 127 (1998), 193.
doi: 10.1007/BF01170373. |
[11] |
H.-H. Dai and Y. Huo, Solitary shock waves and other travelling waves in a general compressible hyperelastic rod,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 456 (2000), 331.
doi: 10.1098/rspa.2000.0520. |
[12] |
K. Grunert, H. Holden and X. Raynaud, Lipschitz metric for the periodic Camassa-Holm equation,, J. Differential Equations, 250 (2011), 1460.
doi: 10.1016/j.jde.2010.07.006. |
[13] |
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.
doi: 10.1080/03605300601088674. |
[14] |
H. Holden and X. Raynaud, Global conservative multipeakon solutions of the Camassa-Holm equation,, J. Hyperbolic Differ. Equ., 4 (2007), 39.
doi: 10.1142/S0219891607001045. |
[15] |
H. Holden and X. Raynaud, Global conservative solutions of the generalized hyperelastic-rod wave equation,, J. Differential Equations, 233 (2007), 448.
doi: 10.1016/j.jde.2006.09.007. |
show all references
References:
[1] |
A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation,, Arch. Ration. Mech. Anal., 183 (2007), 215.
doi: 10.1007/s00205-006-0010-z. |
[2] |
A. Bressan, H. Holden and X. Raynaud, Lipschitz metric for the Hunter-Saxton equation,, J. Math. Pures Appl., 94 (2010), 68.
doi: 10.1016/j.matpur.2010.02.005. |
[3] |
R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solutions,, Phys. Rev. Lett., 71 (1993), 1661.
doi: 10.1103/PhysRevLett.71.1661. |
[4] |
R. Camassa, D. D. Holm and J. Hyman, A new integrable shallow water equation,, Adv. Appl. Mech., 31 (1994), 1. Google Scholar |
[5] |
A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 303.
|
[6] |
A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181 (1998), 229.
doi: 10.1007/BF02392586. |
[7] |
A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation,, Math. Z., 233 (2000), 75.
doi: 10.1007/PL00004793. |
[8] |
A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Rat. Mech. Anal., 192 (2009), 165.
doi: 10.1007/s00205-008-0128-2. |
[9] |
H.-H. Dai, Exact traveling-wave solutions of an integrable equation arising in hyperelastic rods,, Wave Motion, 28 (1998), 367.
doi: 10.1016/S0165-2125(98)00014-6. |
[10] |
H.-H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod,, Acta Mech., 127 (1998), 193.
doi: 10.1007/BF01170373. |
[11] |
H.-H. Dai and Y. Huo, Solitary shock waves and other travelling waves in a general compressible hyperelastic rod,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 456 (2000), 331.
doi: 10.1098/rspa.2000.0520. |
[12] |
K. Grunert, H. Holden and X. Raynaud, Lipschitz metric for the periodic Camassa-Holm equation,, J. Differential Equations, 250 (2011), 1460.
doi: 10.1016/j.jde.2010.07.006. |
[13] |
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.
doi: 10.1080/03605300601088674. |
[14] |
H. Holden and X. Raynaud, Global conservative multipeakon solutions of the Camassa-Holm equation,, J. Hyperbolic Differ. Equ., 4 (2007), 39.
doi: 10.1142/S0219891607001045. |
[15] |
H. Holden and X. Raynaud, Global conservative solutions of the generalized hyperelastic-rod wave equation,, J. Differential Equations, 233 (2007), 448.
doi: 10.1016/j.jde.2006.09.007. |
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