# American Institute of Mathematical Sciences

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June  2017, 37(6): 3285-3299. doi: 10.3934/dcds.2017139

## Local well-posedness of the Camassa-Holm equation on the real line

 1 Department of Mathematics, The Graduate Center, City University of New York, New York, NY 10016, USA 2 Department of Mathematics, Brooklyn College, City University of New York, Brooklyn, NY 11210, USA

Received  December 2016 Revised  January 2017 Published  February 2017

In this paper we prove the local well-posedness of the Camassa-Holm equation on the real line in the space of continuously differentiable diffeomorphisms with an appropriate decaying condition. This work was motivated by G. Misiolek who proved the same result for the Camassa-Holm equation on the periodic domain. We use the Lagrangian approach and rewrite the equation as an ODE on the Banach space. Then by using the standard ODE technique, we prove existence and uniqueness. Finally, we show the continuous dependence of the solution on the initial data by using the topological group property of the diffeomorphism group.

Citation: 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
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##### References:
 [1] Xi Tu, Zhaoyang Yin. Local well-posedness and blow-up phenomena for a generalized Camassa-Holm equation with peakon solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2781-2801. doi: 10.3934/dcds.2016.36.2781 [2] Zhaoyang Yin. Well-posedness and blow-up phenomena for the periodic generalized Camassa-Holm equation. Communications on Pure & Applied Analysis, 2004, 3 (3) : 501-508. doi: 10.3934/cpaa.2004.3.501 [3] Joachim Escher, Olaf Lechtenfeld, Zhaoyang Yin. Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 493-513. doi: 10.3934/dcds.2007.19.493 [4] Jinlu Li, Zhaoyang Yin. Well-posedness and blow-up phenomena for a generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5493-5508. doi: 10.3934/dcds.2016042 [5] Wei Luo, Zhaoyang Yin. Local well-posedness in the critical Besov space and persistence properties for a three-component Camassa-Holm system with N-peakon solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5047-5066. doi: 10.3934/dcds.2016019 [6] Ying Fu, Changzheng Qu, Yichen Ma. Well-posedness and blow-up phenomena for the interacting system of the Camassa-Holm and Degasperis-Procesi equations. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1025-1035. doi: 10.3934/dcds.2010.27.1025 [7] Kai Yan, Zhaoyang Yin. Well-posedness for a modified two-component Camassa-Holm system in critical spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1699-1712. doi: 10.3934/dcds.2013.33.1699 [8] Lei Zhang, Bin Liu. Well-posedness, blow-up criteria and gevrey regularity for a rotation-two-component camassa-holm system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2655-2685. doi: 10.3934/dcds.2018112 [9] Boris Kolev. Local well-posedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167-189. doi: 10.3934/jgm.2017007 [10] 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 [11] 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 [12] 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 [13] 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 [14] 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 [15] 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 [16] Yongsheng Mi, Boling Guo, Chunlai Mu. Persistence properties for the generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019243 [17] Hartmut Pecher. Local well-posedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure & Applied Analysis, 2014, 13 (2) : 673-685. doi: 10.3934/cpaa.2014.13.673 [18] Yongye Zhao, Yongsheng Li, Wei Yan. Local Well-posedness and Persistence Property for the Generalized Novikov Equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 803-820. doi: 10.3934/dcds.2014.34.803 [19] 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 [20] 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

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