# American Institute of Mathematical Sciences

June  2014, 7(2): 305-339. doi: 10.3934/krm.2014.7.305

## On a three-Component Camassa-Holm equation with peakons

 1 College of Mathematics and and Statistics, Chongqing University, Chongqing, 401331, China 2 College of Mathematics and and Statistics, Chongqing University, Chongqing 401331, China

Received  March 2013 Revised  January 2014 Published  March 2014

In this paper, we are concerned with three-Component Camassa-Holm equation with peakons. First, We establish the local well-posedness in a range of the Besov spaces $B^{s}_{p,r},p,r\in [1,\infty],s>\mathrm{ max}\{\frac{3}{2},1+\frac{1}{p}\}$ (which generalize the Sobolev spaces $H^{s}$) by using Littlewood-Paley decomposition and transport equation theory. Second, the local well-posedness in critical case (with $s=\frac{3}{2}, p=2,r=1$) is considered. Then, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. Finally, we consider the initial boundary value problem, our approach is based on sharp extension results for functions on the half-line and several symmetry preserving properties of the equations under discussion.
Citation: Yongsheng Mi, Chunlai Mu. On a three-Component Camassa-Holm equation with peakons. Kinetic & Related Models, 2014, 7 (2) : 305-339. doi: 10.3934/krm.2014.7.305
##### References:
 [1] M. Baouendi and C. Goulaouic, Remarks on the abstract form of nonlinear Cauchy-Kowalevski theorems,, Commun. Partial Differential Equation, 2 (1977), 1151. doi: 10.1080/03605307708820057. Google Scholar [2] M. Baouendi and C. Goulaouic, Sharp estimates for analytic pseudodifferential operators and application to the Cauchy problems,, J. Differential Equations, 48 (1983), 241. doi: 10.1016/0022-0396(83)90051-7. Google Scholar [3] 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 [4] A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation,, Anal. Appl., 5 (2007), 1. doi: 10.1142/S0219530507000857. Google Scholar [5] R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661. Google Scholar [6] J. Chemin, Localization in Fourier space and Navier-Stokes system,, Phase Space Analysis of Partial Differential Equations. Proceedings, (2004), 53. Google Scholar [7] A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach,, Ann. Inst. Fourier (Grenoble), 50 (2000), 321. doi: 10.5802/aif.1757. Google Scholar [8] A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523. doi: 10.1007/s00222-006-0002-5. Google Scholar [9] A. Constantin, On the inverse spectral problem for the Camassa-Holm equation,, J. Funct. Anal., 155 (1998), 352. doi: 10.1006/jfan.1997.3231. Google Scholar [10] A. Constantin and J. Escher, Particle trajectories in solitary water waves,, Bull. Amer. Math. Soc., 44 (2007), 423. doi: 10.1090/S0273-0979-07-01159-7. Google Scholar [11] A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. of Math., 173 (2011), 559. doi: 10.4007/annals.2011.173.1.12. Google Scholar [12] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181 (1998), 229. doi: 10.1007/BF02392586. Google Scholar [13] A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation,, Math. Z., 233 (2000), 75. doi: 10.1007/PL00004793. Google Scholar [14] A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation,, Ann. Scuola Norm. Sup. Pisa., 26 (1998), 303. Google Scholar [15] A. Constantin, V. Gerdjikov and R. Ivanov, Inverse scattering transform for the Camassa-Holm equation,, Inverse Problems, 22 (2006), 2197. doi: 10.1088/0266-5611/22/6/017. Google Scholar [16] A. Constantin and R. Ivanov, On an integrable two-component Camassa-Holm shallow water system,, Phys. Lett. A., 372 (2008), 7129. doi: 10.1016/j.physleta.2008.10.050. Google Scholar [17] A. Constantin, T. Kappeler, B. Kolev and P. Topalov, On geodesic exponential maps of the Virasoro group,, Ann. Global Anal. Geom., 31 (2007), 155. doi: 10.1007/s10455-006-9042-8. Google Scholar [18] A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle,, Commentarii Mathematici Helvetici, 78 (2003), 787. doi: 10.1007/s00014-003-0785-6. Google Scholar [19] 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 [20] A. Constantin and H. P. McKean, A shallow water equation on the circle,, Comm. Pure Appl. Math., 52 (1999), 949. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D. Google Scholar [21] A. Constantin and W. Strauss, Stability of peakons,, Comm. Pure Appl. Math., 53 (2000), 603. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. Google Scholar [22] R. Danchin, A few remarks on the Camassa-Holm equation,, Differential Integral Equations, 14 (2001), 953. Google Scholar [23] R. Danchin, Fourier Analysis Methods for PDEs,, Lecture Notes, (2003). Google Scholar [24] R. Dachin, A note on well-posedness for Camassa-Holm equation,, J. Differential Equations, 192 (2003), 429. doi: 10.1016/S0022-0396(03)00096-2. Google Scholar [25] J. Escher and Z. Yin, Initial boundary value problems for nonlinear dispersive wave equations,, J. Funct. Anal., 256 (2009), 479. doi: 10.1016/j.jfa.2008.07.010. Google Scholar [26] J. Escher and Z. Yin, Initial boundary value problems of the Camassa-Holm equation,, Commun. Partial Differential Equation, 33 (2008), 377. doi: 10.1080/03605300701318872. Google Scholar [27] Y. Fu, G. Gui, Y. Liu and C. Qu, On the Cauchy problem for the integrable Camassa-Holm type equation with cubic nonlinearity,, , (). Google Scholar [28] Y. Fu, Y. Liu and C. Qu, Well-posedness and blow-up solution for a modified twocomponent periodic Camassa-Holm system with peakons,, Math. Ann., 348 (2010), 415. doi: 10.1007/s00208-010-0483-9. Google Scholar [29] 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), 1. doi: 10.1063/1.3064810. Google Scholar [30] Y. Fu and C. Qu, On a new Three-Component Camassa-Holm equation with peakons,, Comm. Theor. Phys., 53 (2010), 223. Google Scholar [31] 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 [32] 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 [33] A. Himonas and G. Misiolek, Analyticity of the Cauchy problem for an integrable evolution equation,, Math. Ann., 327 (2003), 575. doi: 10.1007/s00208-003-0466-1. Google Scholar [34] H. Holden and X. Raynaud, Global conservative solutions of the Camassa-Holm equations-a Lagrangianpoiny of view,, Comm. Partial Differential Equations, 32 (2007), 1511. doi: 10.1080/03605300601088674. Google Scholar [35] 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 [36] Q. Hu, L. Lin and J. Jin, Well-posedness and blow-up phenomena for a new three-component Camassa-Holm system with peakons,, J. Hyper. Differential Equations, 9 (2012), 451. doi: 10.1142/S0219891612500142. Google Scholar [37] D. Holm and R. Ivanov, Multi-component generalizations of the CH equation: geometrical aspects, peakons and numerical examples,, J. Phys. A, 43 (2010), 1. doi: 10.1088/1751-8113/43/49/492001. Google Scholar [38] D. Holm and R. Ivanov, Two-component CH system: inverse scattering, peakons and geometry,, Inverse Problems, 27 (2011), 1. doi: 10.1088/0266-5611/27/4/045013. Google Scholar [39] D. Holm, L. Onaraigh and C. Tronci, Singular solutions of a modified two-component Camassa-Holm equation,, Phys. Rev. E., 79 (2009), 1. doi: 10.1103/PhysRevE.79.016601. Google Scholar [40] S. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group,, J. Math. Phys., 40 (1999), 857. doi: 10.1063/1.532690. Google Scholar [41] J. Lenells, A variational approach to the stability of periodic peakons,, J. Nonlinear Math. Phys., 11 (2004), 151. doi: 10.2991/jnmp.2004.11.2.2. Google Scholar [42] Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation,, J. Differential Equations, 162 (2000), 27. doi: 10.1006/jdeq.1999.3683. Google Scholar [43] G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Virasoro group,, J. Geom. Phys., 24 (1998), 203. doi: 10.1016/S0393-0440(97)00010-7. Google Scholar [44] L. Tian, Y. Wang and J. Zhou, Global conservative and dissipative solutions of a coupled Camassa-Holm equations,, J. Math. Phys., 52 (2011). doi: 10.1063/1.3600216. Google Scholar [45] L. Tian and Y. Xu, Attractor for a viscous coupled Camassa-Holm equation,, Adv. Differ. Equ., (2010). Google Scholar [46] J. F. Toland, Stokes waves,, Topol. Methods. Nonlinear Anal., 7 (1996), 1. Google Scholar [47] K. Yan and Z. Yin, Analytic solutions of the Cauchy problem for two-component shallow water systems,, Math. Z., 269 (2011), 1113. doi: 10.1007/s00209-010-0775-5. Google Scholar [48] M. Zhu and Blow-up, Global Existence and Persistence Properties for the Coupled Camassa-Holm equations,, Math Phys. Anal Geom., 14 (2011), 197. doi: 10.1007/s11040-011-9094-2. Google Scholar

show all references

##### References:
 [1] M. Baouendi and C. Goulaouic, Remarks on the abstract form of nonlinear Cauchy-Kowalevski theorems,, Commun. Partial Differential Equation, 2 (1977), 1151. doi: 10.1080/03605307708820057. Google Scholar [2] M. Baouendi and C. Goulaouic, Sharp estimates for analytic pseudodifferential operators and application to the Cauchy problems,, J. Differential Equations, 48 (1983), 241. doi: 10.1016/0022-0396(83)90051-7. Google Scholar [3] 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 [4] A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation,, Anal. Appl., 5 (2007), 1. doi: 10.1142/S0219530507000857. Google Scholar [5] R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661. Google Scholar [6] J. Chemin, Localization in Fourier space and Navier-Stokes system,, Phase Space Analysis of Partial Differential Equations. Proceedings, (2004), 53. Google Scholar [7] A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach,, Ann. Inst. Fourier (Grenoble), 50 (2000), 321. doi: 10.5802/aif.1757. Google Scholar [8] A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523. doi: 10.1007/s00222-006-0002-5. Google Scholar [9] A. Constantin, On the inverse spectral problem for the Camassa-Holm equation,, J. Funct. Anal., 155 (1998), 352. doi: 10.1006/jfan.1997.3231. Google Scholar [10] A. Constantin and J. Escher, Particle trajectories in solitary water waves,, Bull. Amer. Math. Soc., 44 (2007), 423. doi: 10.1090/S0273-0979-07-01159-7. Google Scholar [11] A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. of Math., 173 (2011), 559. doi: 10.4007/annals.2011.173.1.12. Google Scholar [12] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181 (1998), 229. doi: 10.1007/BF02392586. Google Scholar [13] A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation,, Math. Z., 233 (2000), 75. doi: 10.1007/PL00004793. Google Scholar [14] A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation,, Ann. Scuola Norm. Sup. Pisa., 26 (1998), 303. Google Scholar [15] A. Constantin, V. Gerdjikov and R. Ivanov, Inverse scattering transform for the Camassa-Holm equation,, Inverse Problems, 22 (2006), 2197. doi: 10.1088/0266-5611/22/6/017. Google Scholar [16] A. Constantin and R. Ivanov, On an integrable two-component Camassa-Holm shallow water system,, Phys. Lett. A., 372 (2008), 7129. doi: 10.1016/j.physleta.2008.10.050. Google Scholar [17] A. Constantin, T. Kappeler, B. Kolev and P. Topalov, On geodesic exponential maps of the Virasoro group,, Ann. Global Anal. Geom., 31 (2007), 155. doi: 10.1007/s10455-006-9042-8. Google Scholar [18] A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle,, Commentarii Mathematici Helvetici, 78 (2003), 787. doi: 10.1007/s00014-003-0785-6. Google Scholar [19] 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 [20] A. Constantin and H. P. McKean, A shallow water equation on the circle,, Comm. Pure Appl. Math., 52 (1999), 949. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D. Google Scholar [21] A. Constantin and W. Strauss, Stability of peakons,, Comm. Pure Appl. Math., 53 (2000), 603. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. Google Scholar [22] R. Danchin, A few remarks on the Camassa-Holm equation,, Differential Integral Equations, 14 (2001), 953. Google Scholar [23] R. Danchin, Fourier Analysis Methods for PDEs,, Lecture Notes, (2003). Google Scholar [24] R. Dachin, A note on well-posedness for Camassa-Holm equation,, J. Differential Equations, 192 (2003), 429. doi: 10.1016/S0022-0396(03)00096-2. Google Scholar [25] J. Escher and Z. Yin, Initial boundary value problems for nonlinear dispersive wave equations,, J. Funct. Anal., 256 (2009), 479. doi: 10.1016/j.jfa.2008.07.010. Google Scholar [26] J. Escher and Z. Yin, Initial boundary value problems of the Camassa-Holm equation,, Commun. Partial Differential Equation, 33 (2008), 377. doi: 10.1080/03605300701318872. Google Scholar [27] Y. Fu, G. Gui, Y. Liu and C. Qu, On the Cauchy problem for the integrable Camassa-Holm type equation with cubic nonlinearity,, , (). Google Scholar [28] Y. Fu, Y. Liu and C. Qu, Well-posedness and blow-up solution for a modified twocomponent periodic Camassa-Holm system with peakons,, Math. Ann., 348 (2010), 415. doi: 10.1007/s00208-010-0483-9. Google Scholar [29] 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), 1. doi: 10.1063/1.3064810. Google Scholar [30] Y. Fu and C. Qu, On a new Three-Component Camassa-Holm equation with peakons,, Comm. Theor. Phys., 53 (2010), 223. Google Scholar [31] 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 [32] 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 [33] A. Himonas and G. Misiolek, Analyticity of the Cauchy problem for an integrable evolution equation,, Math. Ann., 327 (2003), 575. doi: 10.1007/s00208-003-0466-1. Google Scholar [34] H. Holden and X. Raynaud, Global conservative solutions of the Camassa-Holm equations-a Lagrangianpoiny of view,, Comm. Partial Differential Equations, 32 (2007), 1511. doi: 10.1080/03605300601088674. Google Scholar [35] 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 [36] Q. Hu, L. Lin and J. Jin, Well-posedness and blow-up phenomena for a new three-component Camassa-Holm system with peakons,, J. Hyper. Differential Equations, 9 (2012), 451. doi: 10.1142/S0219891612500142. Google Scholar [37] D. Holm and R. Ivanov, Multi-component generalizations of the CH equation: geometrical aspects, peakons and numerical examples,, J. Phys. A, 43 (2010), 1. doi: 10.1088/1751-8113/43/49/492001. Google Scholar [38] D. Holm and R. Ivanov, Two-component CH system: inverse scattering, peakons and geometry,, Inverse Problems, 27 (2011), 1. doi: 10.1088/0266-5611/27/4/045013. Google Scholar [39] D. Holm, L. Onaraigh and C. Tronci, Singular solutions of a modified two-component Camassa-Holm equation,, Phys. Rev. E., 79 (2009), 1. doi: 10.1103/PhysRevE.79.016601. Google Scholar [40] S. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group,, J. Math. Phys., 40 (1999), 857. doi: 10.1063/1.532690. Google Scholar [41] J. Lenells, A variational approach to the stability of periodic peakons,, J. Nonlinear Math. Phys., 11 (2004), 151. doi: 10.2991/jnmp.2004.11.2.2. Google Scholar [42] Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation,, J. Differential Equations, 162 (2000), 27. doi: 10.1006/jdeq.1999.3683. Google Scholar [43] G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Virasoro group,, J. Geom. Phys., 24 (1998), 203. doi: 10.1016/S0393-0440(97)00010-7. Google Scholar [44] L. Tian, Y. Wang and J. Zhou, Global conservative and dissipative solutions of a coupled Camassa-Holm equations,, J. Math. Phys., 52 (2011). doi: 10.1063/1.3600216. Google Scholar [45] L. Tian and Y. Xu, Attractor for a viscous coupled Camassa-Holm equation,, Adv. Differ. Equ., (2010). Google Scholar [46] J. F. Toland, Stokes waves,, Topol. Methods. Nonlinear Anal., 7 (1996), 1. Google Scholar [47] K. Yan and Z. Yin, Analytic solutions of the Cauchy problem for two-component shallow water systems,, Math. Z., 269 (2011), 1113. doi: 10.1007/s00209-010-0775-5. Google Scholar [48] M. Zhu and Blow-up, Global Existence and Persistence Properties for the Coupled Camassa-Holm equations,, Math Phys. Anal Geom., 14 (2011), 197. doi: 10.1007/s11040-011-9094-2. Google Scholar
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