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:
[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. 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. 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. 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. doi: 10.1103/PhysRevLett.71.1661.

[6]

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

[7]

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

[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.

[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.

[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.

[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.

[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.

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. 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. 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. 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. doi: 10.1103/PhysRevLett.71.1661.

[6]

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

[7]

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

[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.

[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.

[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.

[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.

[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.

[1]

Li Yang, Zeng Rong, Shouming Zhou, Chunlai Mu. Uniqueness of conservative solutions to the generalized Camassa-Holm equation via characteristics. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5205-5220. doi: 10.3934/dcds.2018230

[2]

Shihui Zhu. Existence and uniqueness of global weak solutions of the Camassa-Holm equation with a forcing. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5201-5221. doi: 10.3934/dcds.2016026

[3]

Yongsheng Mi, Boling Guo, Chunlai Mu. On an $N$-Component Camassa-Holm equation with peakons. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1575-1601. doi: 10.3934/dcds.2017065

[4]

Helge Holden, Xavier Raynaud. Dissipative solutions for the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1047-1112. doi: 10.3934/dcds.2009.24.1047

[5]

Zhenhua Guo, Mina Jiang, Zhian Wang, Gao-Feng Zheng. Global weak solutions to the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 883-906. doi: 10.3934/dcds.2008.21.883

[6]

Milena Stanislavova, Atanas Stefanov. Attractors for the viscous Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 159-186. doi: 10.3934/dcds.2007.18.159

[7]

Defu Chen, Yongsheng Li, Wei Yan. On the Cauchy problem for a generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 871-889. doi: 10.3934/dcds.2015.35.871

[8]

Yu Gao, Jian-Guo Liu. The modified Camassa-Holm equation in Lagrangian coordinates. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2545-2592. doi: 10.3934/dcdsb.2018067

[9]

Stephen C. Anco, Elena Recio, María L. Gandarias, María S. Bruzón. A nonlinear generalization of the Camassa-Holm equation with peakon solutions. Conference Publications, 2015, 2015 (special) : 29-37. doi: 10.3934/proc.2015.0029

[10]

Yongsheng Mi, Chunlai Mu. On a three-Component Camassa-Holm equation with peakons. Kinetic & Related Models, 2014, 7 (2) : 305-339. doi: 10.3934/krm.2014.7.305

[11]

Shouming Zhou, Chunlai Mu. Global conservative and dissipative solutions of the generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1713-1739. doi: 10.3934/dcds.2013.33.1713

[12]

Feng Wang, Fengquan Li, Zhijun Qiao. On the Cauchy problem for a higher-order μ-Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4163-4187. doi: 10.3934/dcds.2018181

[13]

Danping Ding, Lixin Tian, Gang Xu. The study on solutions to Camassa-Holm equation with weak dissipation. Communications on Pure & Applied Analysis, 2006, 5 (3) : 483-492. doi: 10.3934/cpaa.2006.5.483

[14]

Priscila Leal da Silva, Igor Leite Freire. An equation unifying both Camassa-Holm and Novikov equations. Conference Publications, 2015, 2015 (special) : 304-311. doi: 10.3934/proc.2015.0304

[15]

Stephen Anco, Daniel Kraus. Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4449-4465. doi: 10.3934/dcds.2018194

[16]

David F. Parker. Higher-order shallow water equations and the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 629-641. doi: 10.3934/dcdsb.2007.7.629

[17]

Shaoyong Lai, Qichang Xie, Yunxi Guo, YongHong Wu. The existence of weak solutions for a generalized Camassa-Holm equation. Communications on Pure & Applied Analysis, 2011, 10 (1) : 45-57. doi: 10.3934/cpaa.2011.10.45

[18]

Jae Min Lee, Stephen C. Preston. Local well-posedness of the Camassa-Holm equation on the real line. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3285-3299. doi: 10.3934/dcds.2017139

[19]

David Henry. Infinite propagation speed for a two component Camassa-Holm equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 597-606. doi: 10.3934/dcdsb.2009.12.597

[20]

Ying Fu. A note on the Cauchy problem of a modified Camassa-Holm equation with cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2011-2039. doi: 10.3934/dcds.2015.35.2011

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]