Uniqueness of conservative solutions to the Camassa-Holm equation via characteristics

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January 2015, 35(1): 25-42. doi: 10.3934/dcds.2015.35.25

Uniqueness of conservative solutions to the Camassa-Holm equation via characteristics

Alberto Bressan ^1, , Geng Chen ^2, and Qingtian Zhang ^1,

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

Received January 2014 Revised January 2014 Published August 2014

Abstract
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The paper provides a direct proof the uniqueness of solutions to the Camassa-Holm equation, based on characteristics. Given a conservative solution $u=u(t,x)$, an equation is introduced which singles out a unique characteristic curve through each initial point. By studying the evolution of the quantities $u$ and $v= 2\arctan u_x$ along each characteristic, it is proved that the Cauchy problem with general initial data $u_0\in H^1(\mathbb{R})$ has a unique solution, globally in time.

Keywords: singularity, characteristic., Camassa-Holm equation, uniqueness, large data.

Mathematics Subject Classification: Primary: 35L65; Secondary: 35L4.

Citation: Alberto Bressan, Geng Chen, Qingtian Zhang. Uniqueness of conservative solutions to the Camassa-Holm equation via characteristics. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 25-42. doi: 10.3934/dcds.2015.35.25

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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