# American Institute of Mathematical Sciences

November  2009, 24(4): 1047-1112. doi: 10.3934/dcds.2009.24.1047

## Dissipative solutions for the Camassa-Holm equation

 1 Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim 2 Department of Mathematical Sciences, Norwegian University of Science and Technology, NO--7491 Trondheim

Received  June 2008 Revised  November 2008 Published  May 2009

We show that the Camassa--Holm equation $u_t-$uxxt+3uux-$2u_xuxx-uuxxx=0 possesses a global continuous semigroup of weak dissipative solutions for initial data$u|_{t=0}$in$H^1$. The result is obtained by introducing a coordinate transformation into Lagrangian coordinates. Stability in terms of$H^1$and$L^\infty$norm is discussed. Citation: 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  [1] Katrin Grunert, Helge Holden, Xavier Raynaud. Lipschitz metric for the Camassa--Holm equation on the line. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2809-2827. doi: 10.3934/dcds.2013.33.2809 [2] Katrin Grunert, Helge Holden, Xavier Raynaud. Global conservative solutions to the Camassa--Holm equation for initial data with nonvanishing asymptotics. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4209-4227. doi: 10.3934/dcds.2012.32.4209 [3] Min Zhu, Shuanghu Zhang. On the blow-up of solutions to the periodic modified integrable Camassa--Holm equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2347-2364. doi: 10.3934/dcds.2016.36.2347 [4] 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 [5] Helge Holden, Xavier Raynaud. A convergent numerical scheme for the Camassa--Holm equation based on multipeakons. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 505-523. doi: 10.3934/dcds.2006.14.505 [6] Xinglong Wu, Boling Guo. Persistence properties and infinite propagation for the modified 2-component Camassa--Holm equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3211-3223. doi: 10.3934/dcds.2013.33.3211 [7] 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 [8] Xinglong Wu. On the Cauchy problem of a three-component Camassa--Holm equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2827-2854. doi: 10.3934/dcds.2016.36.2827 [9] Katrin Grunert. Blow-up for the two-component Camassa--Holm system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2041-2051. doi: 10.3934/dcds.2015.35.2041 [10] 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 [11] 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 [12] 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 [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] 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 [15] 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 [16] 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 [17] Li Yang, Chunlai Mu, Shouming Zhou, Xinyu Tu. The global conservative solutions for the generalized camassa-holm equation. Electronic Research Archive, 2019, 27: 37-67. doi: 10.3934/era.2019009 [18] 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 [19] 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 [20] 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

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