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

Alberto Bressan; Geng Chen; Qingtian Zhang

doi:10.3934/dcds.2015.35.25

Article Contents

2015, Volume 35, Issue 1: 25-42. Doi: 10.3934/dcds.2015.35.25

This issue Previous Article Stability of the rhomboidal symmetric-mass orbit Next Article Smooth stabilizers for measures on the torus

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

1.
Department of Mathematics, Penn State University, University Park, Pa.16802
2.
School of Mathematics, Georgia Institute of Technology, Atlanta, Ga. 30332

Received: December 31, 2013

Revised: December 31, 2013

Published: December 2014

Abstract Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 35L65; Secondary: 35L45.

Citation:

Related Papers

Cited by

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.