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Uniqueness of conservative solutions to the Camassa-Holm equation via characteristics
1. | Department of Mathematics, Penn State University, University Park, Pa.16802, United States |
2. | School of Mathematics, Georgia Institute of Technology, Atlanta, Ga. 30332, United States |
References:
[1] |
A. Bressan, Hyperbolic Systems of Conservation Laws. The One Dimensional Cauchy Problem, Oxford University Press, 2000. |
[2] |
A. Bressan and A. Constantin, Global conservative solutions to the Camassa-Holm equation, Arch. Rat. Mech. Anal., 183 (2007), 215-239.
doi: 10.1007/s00205-006-0010-z. |
[3] |
A. Bressan and A. Constantin, Global dissipative solutions to the Camassa-Holm equation, Analysis and Applications, 5 (2007), 1-27.
doi: 10.1142/S0219530507000857. |
[4] |
A. Bressan and M. Fonte, An optimal transportation metric for solutions of the Camassa-Holm equation, Methods and Applications of Analysis, 12 (2005), 191-220.
doi: 10.4310/MAA.2005.v12.n2.a7. |
[5] |
R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
[6] |
C. Dafermos, Generalized characteristics and the Hunter-Saxton equation, J. Hyperbolic Diff. Equat., 8, (2011) 159-168.
doi: 10.1142/S0219891611002366. |
[7] |
L. C. Evans, Partial Differential Equations, Second edition, American Mathematical Society, Providence, RI, 2010. |
[8] |
K. Grunert, H. Holden and X. Raynaud, Lipschitz metric for the periodic Camassa-Holm equation, J. Differential Equations, 250 (2011), 1460-1492.
doi: 10.1016/j.jde.2010.07.006. |
[9] |
K. Grunert, H. Holden and X. Raynaud, Lipschitz metric for the Camassa-Holm equation on the line, Discrete Contin. Dyn. Syst., 33 (2013), 2809-2827.
doi: 10.3934/dcds.2013.33.2809. |
[10] |
H. Holden and X. Raynaud, Global conservative solutions of the Camassa-Holm equation - a Lagrangian point of view, Comm. Partial Diff. Equat., 32 (2007), 1511-1549.
doi: 10.1080/03605300601088674. |
[11] |
Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433.
doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5. |
[12] |
Z. Xin and P. Zhang, On the uniqueness and large time behavior of the weak solutions to a shallow water equation, Comm. Partial Diff. Equat., 27 (2002), 1815-1844.
doi: 10.1081/PDE-120016129. |
show all references
References:
[1] |
A. Bressan, Hyperbolic Systems of Conservation Laws. The One Dimensional Cauchy Problem, Oxford University Press, 2000. |
[2] |
A. Bressan and A. Constantin, Global conservative solutions to the Camassa-Holm equation, Arch. Rat. Mech. Anal., 183 (2007), 215-239.
doi: 10.1007/s00205-006-0010-z. |
[3] |
A. Bressan and A. Constantin, Global dissipative solutions to the Camassa-Holm equation, Analysis and Applications, 5 (2007), 1-27.
doi: 10.1142/S0219530507000857. |
[4] |
A. Bressan and M. Fonte, An optimal transportation metric for solutions of the Camassa-Holm equation, Methods and Applications of Analysis, 12 (2005), 191-220.
doi: 10.4310/MAA.2005.v12.n2.a7. |
[5] |
R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
[6] |
C. Dafermos, Generalized characteristics and the Hunter-Saxton equation, J. Hyperbolic Diff. Equat., 8, (2011) 159-168.
doi: 10.1142/S0219891611002366. |
[7] |
L. C. Evans, Partial Differential Equations, Second edition, American Mathematical Society, Providence, RI, 2010. |
[8] |
K. Grunert, H. Holden and X. Raynaud, Lipschitz metric for the periodic Camassa-Holm equation, J. Differential Equations, 250 (2011), 1460-1492.
doi: 10.1016/j.jde.2010.07.006. |
[9] |
K. Grunert, H. Holden and X. Raynaud, Lipschitz metric for the Camassa-Holm equation on the line, Discrete Contin. Dyn. Syst., 33 (2013), 2809-2827.
doi: 10.3934/dcds.2013.33.2809. |
[10] |
H. Holden and X. Raynaud, Global conservative solutions of the Camassa-Holm equation - a Lagrangian point of view, Comm. Partial Diff. Equat., 32 (2007), 1511-1549.
doi: 10.1080/03605300601088674. |
[11] |
Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433.
doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5. |
[12] |
Z. Xin and P. Zhang, On the uniqueness and large time behavior of the weak solutions to a shallow water equation, Comm. Partial Diff. Equat., 27 (2002), 1815-1844.
doi: 10.1081/PDE-120016129. |
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