# American Institute of Mathematical Sciences

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

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

 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 $\rho_0\in L^2(\mathbb{R})$.
Citation: 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
##### References:
 [1] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems,, Clarendon Press, (2000). Google Scholar [2] A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation,, Arch. Ration. Mech. Anal., 183 (2007), 215. doi: 10.1007/s00205-006-0010-z. Google Scholar [3] A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation,, Analysis and Applications, 5 (2007), 1. doi: 10.1142/S0219530507000857. Google Scholar [4] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661. Google Scholar [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. doi: 10.1093/imrn/rnq118. Google Scholar [6] A. Constantin and R. I. Ivanov, On an integrable two-component Camassa-Holm shallow water system,, Physics Letters A, 372 (2008), 7129. doi: 10.1016/j.physleta.2008.10.050. Google Scholar [7] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Ration. Mech. Anal., 192 (2009), 165. doi: 10.1007/s00205-008-0128-2. Google Scholar [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. doi: 10.3934/dcds.2007.19.493. Google Scholar [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). doi: 10.1063/1.3064810. Google Scholar [10] K. Grunert, H. Holden and X. Raynaud, Global solutions for the two-component Camassa-Holm system,, Comm. Partial Differential Equations, 37 (2012), 2245. doi: 10.1080/03605302.2012.683505. Google Scholar [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. doi: 10.1016/j.nonrwa.2013.12.001. Google Scholar [12] K. Grunert, H. Holden and X. Raynaud, A continuous interpolation between conservative and dissipative solutions for the Camassa-Holm system,, , (). Google Scholar [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), 526 (2010), 199. doi: 10.1090/conm/526/10382. Google Scholar [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. doi: 10.1016/j.anihpc.2011.04.003. Google Scholar [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. doi: 10.1016/j.jde.2009.08.002. Google Scholar [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. doi: 10.1016/j.jfa.2010.02.008. Google Scholar [17] G. Gui and Y. Liu, On the Cauchy problem for the two-component Camassa-Holm system,, Math. Z., 268 (2011), 45. doi: 10.1007/s00209-009-0660-2. Google Scholar [18] Z. Guo and Y. Zhou, On solutions to a two-component generalized Camassa-Holm equation,, Stud. Appl. Math., 124 (2010), 307. doi: 10.1111/j.1467-9590.2009.00472.x. Google Scholar [19] D. Henry, Infnite propagation speed for a two-component Camassa-Holm equation,, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 597. doi: 10.3934/dcdsb.2009.12.597. Google Scholar [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. doi: 10.1080/03605300601088674. Google Scholar [21] H. Holden and X. Raynaud, Dissipative solutions for the Camassa-Holm equation,, Discrete Contin. Dyn. Syst., 24 (2009), 1047. doi: 10.3934/dcds.2009.24.1047. Google Scholar [22] Q. Hu, Global existence and blow-up phenomena for a weakly dissipative 2-component Camassa-Holm system,, Applicable Analysis, 92 (2013), 398. doi: 10.1080/00036811.2011.621893. Google Scholar [23] P. A. Kuz'min, Two-component generalizations of the Camassa-Holm equation,, Math. Notes, 81 (2007), 130. doi: 10.1134/S0001434607010142. Google Scholar [24] W. Tan and Z. Yin, Global dissipative solutions of a modified two-component Camassa-Holm shallow water system,, J. Math. Phys., 52 (2011). doi: 10.1063/1.3562928. Google Scholar [25] M. Yuen, Perturbational blowup solutions to the 2-component Camassa-Holm equations,, J. Math. Anal. Appl., 390 (2012), 596. doi: 10.1016/j.jmaa.2011.05.016. Google Scholar [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. doi: 10.1093/imrn/rnp211. Google Scholar

show all references

##### References:
 [1] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems,, Clarendon Press, (2000). Google Scholar [2] A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation,, Arch. Ration. Mech. Anal., 183 (2007), 215. doi: 10.1007/s00205-006-0010-z. Google Scholar [3] A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation,, Analysis and Applications, 5 (2007), 1. doi: 10.1142/S0219530507000857. Google Scholar [4] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661. Google Scholar [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. doi: 10.1093/imrn/rnq118. Google Scholar [6] A. Constantin and R. I. Ivanov, On an integrable two-component Camassa-Holm shallow water system,, Physics Letters A, 372 (2008), 7129. doi: 10.1016/j.physleta.2008.10.050. Google Scholar [7] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Ration. Mech. Anal., 192 (2009), 165. doi: 10.1007/s00205-008-0128-2. Google Scholar [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. doi: 10.3934/dcds.2007.19.493. Google Scholar [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). doi: 10.1063/1.3064810. Google Scholar [10] K. Grunert, H. Holden and X. Raynaud, Global solutions for the two-component Camassa-Holm system,, Comm. Partial Differential Equations, 37 (2012), 2245. doi: 10.1080/03605302.2012.683505. Google Scholar [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. doi: 10.1016/j.nonrwa.2013.12.001. Google Scholar [12] K. Grunert, H. Holden and X. Raynaud, A continuous interpolation between conservative and dissipative solutions for the Camassa-Holm system,, , (). Google Scholar [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), 526 (2010), 199. doi: 10.1090/conm/526/10382. Google Scholar [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. doi: 10.1016/j.anihpc.2011.04.003. Google Scholar [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. doi: 10.1016/j.jde.2009.08.002. Google Scholar [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. doi: 10.1016/j.jfa.2010.02.008. Google Scholar [17] G. Gui and Y. Liu, On the Cauchy problem for the two-component Camassa-Holm system,, Math. Z., 268 (2011), 45. doi: 10.1007/s00209-009-0660-2. Google Scholar [18] Z. Guo and Y. Zhou, On solutions to a two-component generalized Camassa-Holm equation,, Stud. Appl. Math., 124 (2010), 307. doi: 10.1111/j.1467-9590.2009.00472.x. Google Scholar [19] D. Henry, Infnite propagation speed for a two-component Camassa-Holm equation,, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 597. doi: 10.3934/dcdsb.2009.12.597. Google Scholar [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. doi: 10.1080/03605300601088674. Google Scholar [21] H. Holden and X. Raynaud, Dissipative solutions for the Camassa-Holm equation,, Discrete Contin. Dyn. Syst., 24 (2009), 1047. doi: 10.3934/dcds.2009.24.1047. Google Scholar [22] Q. Hu, Global existence and blow-up phenomena for a weakly dissipative 2-component Camassa-Holm system,, Applicable Analysis, 92 (2013), 398. doi: 10.1080/00036811.2011.621893. Google Scholar [23] P. A. Kuz'min, Two-component generalizations of the Camassa-Holm equation,, Math. Notes, 81 (2007), 130. doi: 10.1134/S0001434607010142. Google Scholar [24] W. Tan and Z. Yin, Global dissipative solutions of a modified two-component Camassa-Holm shallow water system,, J. Math. Phys., 52 (2011). doi: 10.1063/1.3562928. Google Scholar [25] M. Yuen, Perturbational blowup solutions to the 2-component Camassa-Holm equations,, J. Math. Anal. Appl., 390 (2012), 596. doi: 10.1016/j.jmaa.2011.05.016. Google Scholar [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. doi: 10.1093/imrn/rnp211. Google Scholar
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