American Institute of Mathematical Sciences

Advanced
September  2007, 19(3): 493-513. doi: 10.3934/dcds.2007.19.493

Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation

Joachim Escher 1, , Olaf Lechtenfeld 2, and  Zhaoyang Yin 3,

1. 

Institute for Applied Mathematics, Leibniz University of Hanover, D-30167 Hanover, Germany
2. 

Institute for Theoretical Physics, Leibniz University of Hanover, Appelstraße 2, D-30167 Hanover, Germany
3. 

Department of Mathematics, Sun Yat-sen University, 510275 Guangzhou, China

Received  April 2007 Revised  June 2007 Published  July 2007

After some remarks on a possible zero-curvature formulation we first establish local well-posedness for the 2-component Camassa-Holm equation. Then precise blow-up scenarios for strong solutions to the system are derived. Finally we present two blow-up results for strong solutions to the system.
Keywords: blow-up phenomena., strong solutions, local well-posedness, The 2-component Camass-Holm equation.
Mathematics Subject Classification: 35G25, 35L0.
Citation: Joachim Escher, Olaf Lechtenfeld, Zhaoyang Yin. Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2007, 19 (3) : 493-513. doi: 10.3934/dcds.2007.19.493
[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, 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]

Jinlu Li, Zhaoyang Yin. Well-posedness and blow-up phenomena for a generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5493-5508. doi: 10.3934/dcds.2016042
[4]

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, 2010, 27 (3) : 1025-1035. doi: 10.3934/dcds.2010.27.1025
[5]

Lei Zhang, Bin Liu. Well-posedness, blow-up criteria and gevrey regularity for a rotation-two-component camassa-holm system. Discrete & Continuous Dynamical Systems, 2018, 38 (5) : 2655-2685. doi: 10.3934/dcds.2018112
[6]

Yongsheng Mi, Boling Guo, Chunlai Mu. Well-posedness and blow-up scenario for a new integrable four-component system with peakon solutions. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 2171-2191. doi: 10.3934/dcds.2016.36.2171
[7]

Luigi Forcella, Kazumasa Fujiwara, Vladimir Georgiev, Tohru Ozawa. Local well-posedness and blow-up for the half Ginzburg-Landau-Kuramoto equation with rough coefficients and potential. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2661-2678. doi: 10.3934/dcds.2019111
[8]

Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843
[9]

Xinwei Yu, Zhichun Zhai. On the Lagrangian averaged Euler equations: local well-posedness and blow-up criterion. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1809-1823. doi: 10.3934/cpaa.2012.11.1809
[10]

Wenjing Zhao. Local well-posedness and blow-up criteria of magneto-viscoelastic flows. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4637-4655. doi: 10.3934/dcds.2018203
[11]

Vo Van Au, Jagdev Singh, Anh Tuan Nguyen. Well-posedness results and blow-up for a semi-linear time fractional diffusion equation with variable coefficients. Electronic Research Archive, , () : -. doi: 10.3934/era.2021052
[12]

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, 2016, 36 (9) : 5047-5066. doi: 10.3934/dcds.2016019
[13]

Jae Min Lee, Stephen C. Preston. Local well-posedness of the Camassa-Holm equation on the real line. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3285-3299. doi: 10.3934/dcds.2017139
[14]

Wenxia Chen, Jingyi Liu, Danping Ding, Lixin Tian. Blow-up for two-component Camassa-Holm equation with generalized weak dissipation. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3769-3784. doi: 10.3934/cpaa.2020166
[15]

Katrin Grunert. Blow-up for the two-component Camassa--Holm system. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 2041-2051. doi: 10.3934/dcds.2015.35.2041
[16]

Xinglong Wu, Boling Guo. Persistence properties and infinite propagation for the modified 2-component Camassa--Holm equation. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 3211-3223. doi: 10.3934/dcds.2013.33.3211
[17]

Tarek Saanouni. A note on global well-posedness and blow-up of some semilinear evolution equations. Evolution Equations & Control Theory, 2015, 4 (3) : 355-372. doi: 10.3934/eect.2015.4.355
[18]

Min Li, Zhaoyang Yin. Blow-up phenomena and travelling wave solutions to the periodic integrable dispersive Hunter-Saxton equation. Discrete & Continuous Dynamical Systems, 2017, 37 (12) : 6471-6485. doi: 10.3934/dcds.2017280
[19]

Min Zhu, Shuanghu Zhang. Blow-up of solutions to the periodic modified Camassa-Holm equation with varying linear dispersion. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 7235-7256. doi: 10.3934/dcds.2016115
[20]

Min Zhu, Ying Wang. Blow-up of solutions to the periodic generalized modified Camassa-Holm equation with varying linear dispersion. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 645-661. doi: 10.3934/dcds.2017027

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (280)
  • HTML views (0)
  • Cited by (202)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences