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May  2015, 35(5): 2041-2051. doi: 10.3934/dcds.2015.35.2041

Blow-up for the two-component Camassa--Holm system

Katrin Grunert 1,

1. 

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

Received  June 2014 Revised  August 2014 Published  December 2014

Following conservative solutions of the two-component Camassa--Holm system $u_t-u_{txx}+3uu_x-2u_xu_{xx}-uu_{xxx}+\rho\rho_x=0$, $\rho_t+(u\rho)_x=0$ along characteristics, we determine if wave breaking occurs in the nearby future or not, for initial data $u_0\in H^1(\mathbb{R})$ and $\rho_0\in L^2(\mathbb{R})$.
Keywords: blow up, Two-component Camassa--Holm system, regularization..
Mathematics Subject Classification: Primary: 35Q53, 35B35; Secondary: 35B4.
Citation: 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
References:
[1]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, New York, 2000.

[2]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z.

[3]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Analysis and Applications, 5 (2007), 1-27. doi: 10.1142/S0219530507000857.

[4]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.

[5]

R. M. Chen and Y. Liu, Wave breaking and global existence for a generalized two-component Camassa-Holm system, Inter. Math Research Notices, (2011), 1381-1416. doi: 10.1093/imrn/rnq118.

[6]

A. Constantin and R. I. Ivanov, On an integrable two-component Camassa-Holm shallow water system, Physics Letters A, 372 (2008), 7129-7132. doi: 10.1016/j.physleta.2008.10.050.

[7]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2.

[8]

J. Escher, O. Lechtenfeld and Z. Yin, Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation, Discrete Contin. Dyn. Syst., 19 (2007), 493-513. doi: 10.3934/dcds.2007.19.493.

[9]

Y. Fu and C. Qu, Well posedness and blow-up solution for a new coupled Camassa-Holm equations with peakons, J. Math. Phys., 50 (2009), 012906, 25pp. doi: 10.1063/1.3064810.

[10]

K. Grunert, H. Holden and X. Raynaud, Global solutions for the two-component Camassa-Holm system, Comm. Partial Differential Equations, 37 (2012), 2245-2271. doi: 10.1080/03605302.2012.683505.

[11]

K. Grunert, H. Holden and X. Raynaud, Global dissipative solutions of the two-component Camassa-Holm system for initial data with nonvanishing asymptotics, Nonlinear Anal. Real World Appl., 17 (2014), 203-244. doi: 10.1016/j.nonrwa.2013.12.001.

[12]

K. Grunert, H. Holden and X. Raynaud, A continuous interpolation between conservative and dissipative solutions for the Camassa-Holm system, arXiv:1402.1060.

[13]

C. Guan, K. H. Karlsen and Z. Yin, Well-posedness and blow-up phenomena for a modified two-component Camassa-Holm equation, in Nonlinear Partial Differential Equations and Hyperbolic Wave Phenomena (eds. H. Holden and K. H. Karlsen), Amer. Math. Soc., Providence, 526 (2010), 199-220. doi: 10.1090/conm/526/10382.

[14]

C. Guan and Z. Yin, Global weak solutions for a modified two-component Camassa-Holm equation, Ann. I. H. Poincaré - AN, 28 (2011), 623-641. doi: 10.1016/j.anihpc.2011.04.003.

[15]

C. Guan and Z. Yin, Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system, J. Differential Equations, 248 (2010), 2003-2014. doi: 10.1016/j.jde.2009.08.002.

[16]

G. Gui and Y. Liu, On the global existence and wave-breaking criteria for the two-component Camassa-Holm system, J. Funct. Anal., 258 (2010), 4251-4278. doi: 10.1016/j.jfa.2010.02.008.

[17]

G. Gui and Y. Liu, On the Cauchy problem for the two-component Camassa-Holm system, Math. Z., 268 (2011), 45-66. doi: 10.1007/s00209-009-0660-2.

[18]

Z. Guo and Y. Zhou, On solutions to a two-component generalized Camassa-Holm equation, Stud. Appl. Math., 124 (2010), 307-322. doi: 10.1111/j.1467-9590.2009.00472.x.

[19]

D. Henry, Infnite propagation speed for a two-component Camassa-Holm equation, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 597-606. doi: 10.3934/dcdsb.2009.12.597.

[20]

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.

[21]

H. Holden and X. Raynaud, Dissipative solutions for the Camassa-Holm equation, Discrete Contin. Dyn. Syst., 24 (2009), 1047-1112. doi: 10.3934/dcds.2009.24.1047.

[22]

Q. Hu, Global existence and blow-up phenomena for a weakly dissipative 2-component Camassa-Holm system, Applicable Analysis, 92 (2013), 398-410. doi: 10.1080/00036811.2011.621893.

[23]

P. A. Kuz'min, Two-component generalizations of the Camassa-Holm equation, Math. Notes, 81 (2007), 130-134. doi: 10.1134/S0001434607010142.

[24]

W. Tan and Z. Yin, Global dissipative solutions of a modified two-component Camassa-Holm shallow water system, J. Math. Phys., 52 (2011), 033507, 24pp. doi: 10.1063/1.3562928.

[25]

M. Yuen, Perturbational blowup solutions to the 2-component Camassa-Holm equations, J. Math. Anal. Appl., 390 (2012), 596-602. doi: 10.1016/j.jmaa.2011.05.016.

[26]

P. Zhang and Y. Liu, Stability of solitary waves and wave-breaking phenomena for the two-component Camassa-Holm system, Int. Math. Res. Not. IMRN, 11 (2010), 1981-2021. doi: 10.1093/imrn/rnp211.

show all references

References:
[1]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, New York, 2000.

[2]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z.

[3]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Analysis and Applications, 5 (2007), 1-27. doi: 10.1142/S0219530507000857.

[4]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.

[5]

R. M. Chen and Y. Liu, Wave breaking and global existence for a generalized two-component Camassa-Holm system, Inter. Math Research Notices, (2011), 1381-1416. doi: 10.1093/imrn/rnq118.

[6]

A. Constantin and R. I. Ivanov, On an integrable two-component Camassa-Holm shallow water system, Physics Letters A, 372 (2008), 7129-7132. doi: 10.1016/j.physleta.2008.10.050.

[7]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2.

[8]

J. Escher, O. Lechtenfeld and Z. Yin, Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation, Discrete Contin. Dyn. Syst., 19 (2007), 493-513. doi: 10.3934/dcds.2007.19.493.

[9]

Y. Fu and C. Qu, Well posedness and blow-up solution for a new coupled Camassa-Holm equations with peakons, J. Math. Phys., 50 (2009), 012906, 25pp. doi: 10.1063/1.3064810.

[10]

K. Grunert, H. Holden and X. Raynaud, Global solutions for the two-component Camassa-Holm system, Comm. Partial Differential Equations, 37 (2012), 2245-2271. doi: 10.1080/03605302.2012.683505.

[11]

K. Grunert, H. Holden and X. Raynaud, Global dissipative solutions of the two-component Camassa-Holm system for initial data with nonvanishing asymptotics, Nonlinear Anal. Real World Appl., 17 (2014), 203-244. doi: 10.1016/j.nonrwa.2013.12.001.

[12]

K. Grunert, H. Holden and X. Raynaud, A continuous interpolation between conservative and dissipative solutions for the Camassa-Holm system, arXiv:1402.1060.

[13]

C. Guan, K. H. Karlsen and Z. Yin, Well-posedness and blow-up phenomena for a modified two-component Camassa-Holm equation, in Nonlinear Partial Differential Equations and Hyperbolic Wave Phenomena (eds. H. Holden and K. H. Karlsen), Amer. Math. Soc., Providence, 526 (2010), 199-220. doi: 10.1090/conm/526/10382.

[14]

C. Guan and Z. Yin, Global weak solutions for a modified two-component Camassa-Holm equation, Ann. I. H. Poincaré - AN, 28 (2011), 623-641. doi: 10.1016/j.anihpc.2011.04.003.

[15]

C. Guan and Z. Yin, Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system, J. Differential Equations, 248 (2010), 2003-2014. doi: 10.1016/j.jde.2009.08.002.

[16]

G. Gui and Y. Liu, On the global existence and wave-breaking criteria for the two-component Camassa-Holm system, J. Funct. Anal., 258 (2010), 4251-4278. doi: 10.1016/j.jfa.2010.02.008.

[17]

G. Gui and Y. Liu, On the Cauchy problem for the two-component Camassa-Holm system, Math. Z., 268 (2011), 45-66. doi: 10.1007/s00209-009-0660-2.

[18]

Z. Guo and Y. Zhou, On solutions to a two-component generalized Camassa-Holm equation, Stud. Appl. Math., 124 (2010), 307-322. doi: 10.1111/j.1467-9590.2009.00472.x.

[19]

D. Henry, Infnite propagation speed for a two-component Camassa-Holm equation, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 597-606. doi: 10.3934/dcdsb.2009.12.597.

[20]

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.

[21]

H. Holden and X. Raynaud, Dissipative solutions for the Camassa-Holm equation, Discrete Contin. Dyn. Syst., 24 (2009), 1047-1112. doi: 10.3934/dcds.2009.24.1047.

[22]

Q. Hu, Global existence and blow-up phenomena for a weakly dissipative 2-component Camassa-Holm system, Applicable Analysis, 92 (2013), 398-410. doi: 10.1080/00036811.2011.621893.

[23]

P. A. Kuz'min, Two-component generalizations of the Camassa-Holm equation, Math. Notes, 81 (2007), 130-134. doi: 10.1134/S0001434607010142.

[24]

W. Tan and Z. Yin, Global dissipative solutions of a modified two-component Camassa-Holm shallow water system, J. Math. Phys., 52 (2011), 033507, 24pp. doi: 10.1063/1.3562928.

[25]

M. Yuen, Perturbational blowup solutions to the 2-component Camassa-Holm equations, J. Math. Anal. Appl., 390 (2012), 596-602. doi: 10.1016/j.jmaa.2011.05.016.

[26]

P. Zhang and Y. Liu, Stability of solitary waves and wave-breaking phenomena for the two-component Camassa-Holm system, Int. Math. Res. Not. IMRN, 11 (2010), 1981-2021. doi: 10.1093/imrn/rnp211.

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