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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

Katrin Grunert 1, , Helge Holden 1, and  Xavier Raynaud 2,

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.
Keywords: continuous semigroup., nonvanishing asymptotics, conservative and global solutions, Camassa--Holm equation.
Mathematics Subject Classification: Primary: 35Q53, 35B30; Secondary: 35B35, 35D3.
Citation: Katrin Grunert, Helge Holden, Xavier Raynaud. Global conservative solutions to the Camassa--Holm equation for initial data with nonvanishing asymptotics. Discrete & 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.  Google Scholar

[2]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239.  Google Scholar

[3]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Analysis and Applications, 5 (2007), 1-27.  Google Scholar

[4]

A. Bressan, H. Holden and X. Raynaud, Lipschitz metric for the Hunter-Saxton equation, J. Math. Pures Appl., 94 (2010), 68-92.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

A. Constantin, Finite propagation speed for the Camassa-Holm equation, J. Math. Phys., 41 (2005), 023506.  Google Scholar

[8]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

K. Grunert, H. Holden and X. Raynaud, Lipschitz metric for the Camassa-Holm equation on the line,, Discrete Contin. Dyn. Syst., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239.  Google Scholar

[3]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Analysis and Applications, 5 (2007), 1-27.  Google Scholar

[4]

A. Bressan, H. Holden and X. Raynaud, Lipschitz metric for the Hunter-Saxton equation, J. Math. Pures Appl., 94 (2010), 68-92.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

A. Constantin, Finite propagation speed for the Camassa-Holm equation, J. Math. Phys., 41 (2005), 023506.  Google Scholar

[8]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

K. Grunert, H. Holden and X. Raynaud, Lipschitz metric for the Camassa-Holm equation on the line,, Discrete Contin. Dyn. Syst., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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