# American Institute of Mathematical Sciences

• Previous Article
Stability and convergence at infinite time of several fully discrete schemes for a Ginzburg-Landau model for nematic liquid crystal flows
• DCDS Home
• This Issue
• Next Article
Entropy-expansiveness for partially hyperbolic diffeomorphisms
December  2012, 32(12): 4209-4227. doi: 10.3934/dcds.2012.32.4209

## Global conservative solutions to the Camassa--Holm equation for initial data with nonvanishing asymptotics

 1 Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway, Norway 2 Centre of Mathematics for Applications, University of Oslo, NO-0316 Oslo, Norway

Received  June 2011 Revised  January 2012 Published  August 2012

We show existence of global conservative solutions of the Cauchy problem for the Camassa--Holm equation $u_t-u_{txx}+\kappa u_x+3uu_x-2u_xu_{xx}-uu_{xxx}=0$ with nonvanishing and distinct spatial asymptotics.
Citation: Katrin Grunert, Helge Holden, Xavier Raynaud. Global conservative solutions to the Camassa--Holm equation for initial data with nonvanishing asymptotics. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4209-4227. doi: 10.3934/dcds.2012.32.4209
##### References:
 [1] M. Bendahmane, G. M. Coclite and K. H. Karlsen, $H^1$-perturbations of smooth solutions for a weakly dissipative hyperelastic-rod wave equation, Mediterranean Journal of Mathematics, 3 (2006), 419-432. [2] A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239. [3] A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Analysis and Applications, 5 (2007), 1-27. [4] A. Bressan, H. Holden and X. Raynaud, Lipschitz metric for the Hunter-Saxton equation, J. Math. Pures Appl., 94 (2010), 68-92. [5] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solutions, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661. [6] R. Camassa, D. D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. doi: 10.1016/S0065-2156(08)70254-0. [7] A. Constantin, Finite propagation speed for the Camassa-Holm equation, J. Math. Phys., 41 (2005), 023506. [8] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586. [9] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Rat. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2. [10] 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. [11] K. Grunert, H. Holden and X. Raynaud, Lipschitz metric for the Camassa-Holm equation on the line,, Discrete Contin. Dyn. Syst., (). [12] H. Holden and X. Raynaud, Global conservative solutions for the Camassa-Holm equation - a Lagrangian point of view, Comm. Partial Differential Equations, 32 (2007), 1511-1549. doi: 10.1080/03605300601088674. [13] H. Holden and X. Raynaud, Global conservative solutions of the generalized hyperelastic-rod wave equation, J. Differential Equations, 233 (2007), 448-484. doi: 10.1016/j.jde.2006.09.007. [14] H. Holden and X. Raynaud, Dissipative solutions of the Camassa-Holm equation, Discrete Contin. Dyn. Syst., 24 (2009), 1047-1112. doi: 10.3934/dcds.2009.24.1047. [15] H. Holden and X. Raynaud, Periodic conservative solutions of the Camassa-Holm equation, Ann. Inst. Fourier (Grenoble), 58 (2008), 945-988. doi: 10.5802/aif.2375. [16] J. Lenells, Traveling wave solutions of the Camassa-Holm equation, J. Differential Equations, 217 (2005), 393-430. doi: 10.1016/j.jde.2004.09.007. [17] J. Lenells, Classification of all traveling-wave solutions for some nonlinear dispersive equations, Phil. Trans. R. Soc. A, 365 (2007), 2291-2298. doi: 10.1098/rsta.2007.2009.

show all references

##### References:
 [1] M. Bendahmane, G. M. Coclite and K. H. Karlsen, $H^1$-perturbations of smooth solutions for a weakly dissipative hyperelastic-rod wave equation, Mediterranean Journal of Mathematics, 3 (2006), 419-432. [2] A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239. [3] A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Analysis and Applications, 5 (2007), 1-27. [4] A. Bressan, H. Holden and X. Raynaud, Lipschitz metric for the Hunter-Saxton equation, J. Math. Pures Appl., 94 (2010), 68-92. [5] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solutions, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661. [6] R. Camassa, D. D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. doi: 10.1016/S0065-2156(08)70254-0. [7] A. Constantin, Finite propagation speed for the Camassa-Holm equation, J. Math. Phys., 41 (2005), 023506. [8] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586. [9] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Rat. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2. [10] 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. [11] K. Grunert, H. Holden and X. Raynaud, Lipschitz metric for the Camassa-Holm equation on the line,, Discrete Contin. Dyn. Syst., (). [12] H. Holden and X. Raynaud, Global conservative solutions for the Camassa-Holm equation - a Lagrangian point of view, Comm. Partial Differential Equations, 32 (2007), 1511-1549. doi: 10.1080/03605300601088674. [13] H. Holden and X. Raynaud, Global conservative solutions of the generalized hyperelastic-rod wave equation, J. Differential Equations, 233 (2007), 448-484. doi: 10.1016/j.jde.2006.09.007. [14] H. Holden and X. Raynaud, Dissipative solutions of the Camassa-Holm equation, Discrete Contin. Dyn. Syst., 24 (2009), 1047-1112. doi: 10.3934/dcds.2009.24.1047. [15] H. Holden and X. Raynaud, Periodic conservative solutions of the Camassa-Holm equation, Ann. Inst. Fourier (Grenoble), 58 (2008), 945-988. doi: 10.5802/aif.2375. [16] J. Lenells, Traveling wave solutions of the Camassa-Holm equation, J. Differential Equations, 217 (2005), 393-430. doi: 10.1016/j.jde.2004.09.007. [17] J. Lenells, Classification of all traveling-wave solutions for some nonlinear dispersive equations, Phil. Trans. R. Soc. A, 365 (2007), 2291-2298. doi: 10.1098/rsta.2007.2009.
 [1] Shouming Zhou, Chunlai Mu. Global conservative and dissipative solutions of the generalized Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1713-1739. doi: 10.3934/dcds.2013.33.1713 [2] 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 [3] Katrin Grunert, Helge Holden, Xavier Raynaud. Lipschitz metric for the Camassa--Holm equation on the line. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2809-2827. doi: 10.3934/dcds.2013.33.2809 [4] Rong Chen, Shihang Pan, Baoshuai Zhang. Global conservative solutions for a modified periodic coupled Camassa-Holm system. Electronic Research Archive, 2021, 29 (1) : 1691-1708. doi: 10.3934/era.2020087 [5] Li Yang, Zeng Rong, Shouming Zhou, Chunlai Mu. Uniqueness of conservative solutions to the generalized Camassa-Holm equation via characteristics. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5205-5220. doi: 10.3934/dcds.2018230 [6] Alberto Bressan, Geng Chen, Qingtian Zhang. Uniqueness of conservative solutions to the Camassa-Holm equation via characteristics. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 25-42. doi: 10.3934/dcds.2015.35.25 [7] Zhenhua Guo, Mina Jiang, Zhian Wang, Gao-Feng Zheng. Global weak solutions to the Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 883-906. doi: 10.3934/dcds.2008.21.883 [8] Min Zhu, Shuanghu Zhang. On the blow-up of solutions to the periodic modified integrable Camassa--Holm equation. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2347-2364. doi: 10.3934/dcds.2016.36.2347 [9] Helge Holden, Xavier Raynaud. A convergent numerical scheme for the Camassa--Holm equation based on multipeakons. Discrete and Continuous Dynamical Systems, 2006, 14 (3) : 505-523. doi: 10.3934/dcds.2006.14.505 [10] Octavian G. Mustafa. Global conservative solutions of the Dullin-Gottwald-Holm equation. Discrete and Continuous Dynamical Systems, 2007, 19 (3) : 575-594. doi: 10.3934/dcds.2007.19.575 [11] Shihui Zhu. Existence and uniqueness of global weak solutions of the Camassa-Holm equation with a forcing. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 5201-5221. doi: 10.3934/dcds.2016026 [12] Xinglong Wu, Boling Guo. Persistence properties and infinite propagation for the modified 2-component Camassa--Holm equation. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 3211-3223. doi: 10.3934/dcds.2013.33.3211 [13] Helge Holden, Xavier Raynaud. Dissipative solutions for the Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1047-1112. doi: 10.3934/dcds.2009.24.1047 [14] Xinglong Wu. On the Cauchy problem of a three-component Camassa--Holm equations. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2827-2854. doi: 10.3934/dcds.2016.36.2827 [15] Katrin Grunert. Blow-up for the two-component Camassa--Holm system. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2041-2051. doi: 10.3934/dcds.2015.35.2041 [16] 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 [17] Danping Ding, Lixin Tian, Gang Xu. The study on solutions to Camassa-Holm equation with weak dissipation. Communications on Pure and Applied Analysis, 2006, 5 (3) : 483-492. doi: 10.3934/cpaa.2006.5.483 [18] Stephen Anco, Daniel Kraus. Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4449-4465. doi: 10.3934/dcds.2018194 [19] Shaoyong Lai, Qichang Xie, Yunxi Guo, YongHong Wu. The existence of weak solutions for a generalized Camassa-Holm equation. Communications on Pure and Applied Analysis, 2011, 10 (1) : 45-57. doi: 10.3934/cpaa.2011.10.45 [20] Yonghui Zhou, Shuguan Ji. Wave breaking phenomena and global existence for the weakly dissipative generalized Camassa-Holm equation. Communications on Pure and Applied Analysis, 2022, 21 (2) : 555-566. doi: 10.3934/cpaa.2021188

2020 Impact Factor: 1.392