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

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

Received  January 2014 Revised  January 2014 Published  August 2014

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.
Citation: Alberto Bressan, Geng Chen, Qingtian Zhang. Uniqueness of conservative solutions to the Camassa-Holm equation via characteristics. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 25-42. doi: 10.3934/dcds.2015.35.25
References:
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A. Bressan and A. Constantin, Global conservative solutions to the Camassa-Holm equation,, Arch. Rat. Mech. Anal., 183 (2007), 215.  doi: 10.1007/s00205-006-0010-z.  Google Scholar

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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.  Google Scholar

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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.  doi: 10.1080/03605300601088674.  Google Scholar

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Z. Xin and P. Zhang, On the weak solutions to a shallow water equation,, Comm. Pure Appl. Math., 53 (2000), 1411.  doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5.  Google Scholar

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show all references

References:
[1]

A. Bressan, Hyperbolic Systems of Conservation Laws. The One Dimensional Cauchy Problem,, Oxford University Press, (2000).   Google Scholar

[2]

A. Bressan and A. Constantin, Global conservative solutions to the Camassa-Holm equation,, Arch. Rat. Mech. Anal., 183 (2007), 215.  doi: 10.1007/s00205-006-0010-z.  Google Scholar

[3]

A. Bressan and A. Constantin, Global dissipative solutions to the Camassa-Holm equation,, Analysis and Applications, 5 (2007), 1.  doi: 10.1142/S0219530507000857.  Google Scholar

[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.  doi: 10.4310/MAA.2005.v12.n2.a7.  Google Scholar

[5]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661.  doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[6]

C. Dafermos, Generalized characteristics and the Hunter-Saxton equation,, J. Hyperbolic Diff. Equat., 8 (2011), 159.  doi: 10.1142/S0219891611002366.  Google Scholar

[7]

L. C. Evans, Partial Differential Equations,, Second edition, (2010).   Google Scholar

[8]

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.  Google Scholar

[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.  doi: 10.3934/dcds.2013.33.2809.  Google Scholar

[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.  doi: 10.1080/03605300601088674.  Google Scholar

[11]

Z. Xin and P. Zhang, On the weak solutions to a shallow water equation,, Comm. Pure Appl. Math., 53 (2000), 1411.  doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5.  Google Scholar

[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.  doi: 10.1081/PDE-120016129.  Google Scholar

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