# 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 and 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), 1151-1162. doi: 10.1080/03605307708820057. [2] M. Baouendi and C. Goulaouic, Sharp estimates for analytic pseudodifferential operators and application to the Cauchy problems, J. Differential Equations, 48 (1983), 241-268. doi: 10.1016/0022-0396(83)90051-7. [3] 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. [4] A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27. doi: 10.1142/S0219530507000857. [5] R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661. [6] J. Chemin, Localization in Fourier space and Navier-Stokes system, Phase Space Analysis of Partial Differential Equations. Proceedings, CRM series, Pisa, (2004), 53-135. [7] A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362. doi: 10.5802/aif.1757. [8] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5. [9] A. Constantin, On the inverse spectral problem for the Camassa-Holm equation, J. Funct. Anal., 155 (1998), 352-363. doi: 10.1006/jfan.1997.3231. [10] A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431. doi: 10.1090/S0273-0979-07-01159-7. [11] A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12. [12] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586. [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), 75-91. doi: 10.1007/PL00004793. [14] A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa., 26 (1998), 303-328. [15] A. Constantin, V. Gerdjikov and R. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207. doi: 10.1088/0266-5611/22/6/017. [16] A. Constantin and R. Ivanov, On an integrable two-component Camassa-Holm shallow water system, Phys. Lett. A., 372 (2008), 7129-7132. doi: 10.1016/j.physleta.2008.10.050. [17] A. Constantin, T. Kappeler, B. Kolev and P. Topalov, On geodesic exponential maps of the Virasoro group, Ann. Global Anal. Geom., 31 (2007), 155-180. doi: 10.1007/s10455-006-9042-8. [18] A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Commentarii Mathematici Helvetici, 78 (2003), 787-804. doi: 10.1007/s00014-003-0785-6. [19] 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. [20] A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D. [21] A. Constantin and W. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. [22] R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988. [23] R. Danchin, Fourier Analysis Methods for PDEs, Lecture Notes, 14 November, 2003. [24] R. Dachin, A note on well-posedness for Camassa-Holm equation, J. Differential Equations, 192 (2003), 429-444. doi: 10.1016/S0022-0396(03)00096-2. [25] J. Escher and Z. Yin, Initial boundary value problems for nonlinear dispersive wave equations, J. Funct. Anal., 256 (2009), 479-508. doi: 10.1016/j.jfa.2008.07.010. [26] J. Escher and Z. Yin, Initial boundary value problems of the Camassa-Holm equation, Commun. Partial Differential Equation, 33 (2008), 377-395. doi: 10.1080/03605300701318872. [27] Y. Fu, G. Gui, Y. Liu and C. Qu, On the Cauchy problem for the integrable Camassa-Holm type equation with cubic nonlinearity, arXiv:1108.5368v2. [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), 415-448. doi: 10.1007/s00208-010-0483-9. [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), 012906, 1-25. doi: 10.1063/1.3064810. [30] Y. Fu and C. Qu, On a new Three-Component Camassa-Holm equation with peakons, Comm. Theor. Phys., 53 (2010), 223-230. [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), 4251-4278. doi: 10.1016/j.jfa.2010.02.008. [32] 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. [33] A. Himonas and G. Misiolek, Analyticity of the Cauchy problem for an integrable evolution equation, Math. Ann., 327 (2003), 575-584. doi: 10.1007/s00208-003-0466-1. [34] H. Holden and X. Raynaud, Global conservative solutions of the Camassa-Holm equations-a Lagrangianpoiny of view, Comm. Partial Differential Equations, 32 (2007), 1511-1549. doi: 10.1080/03605300601088674. [35] 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. [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), 451-467. doi: 10.1142/S0219891612500142. [37] D. Holm and R. Ivanov, Multi-component generalizations of the CH equation: geometrical aspects, peakons and numerical examples, J. Phys. A, 43 (2010), 492001, 1-21. doi: 10.1088/1751-8113/43/49/492001. [38] D. Holm and R. Ivanov, Two-component CH system: inverse scattering, peakons and geometry, Inverse Problems, 27 (2011), 045013, 1-21. doi: 10.1088/0266-5611/27/4/045013. [39] D. Holm, L. Onaraigh and C. Tronci, Singular solutions of a modified two-component Camassa-Holm equation, Phys. Rev. E., 79 (2009), 016601, 1-9. doi: 10.1103/PhysRevE.79.016601. [40] S. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group, J. Math. Phys., 40 (1999), 857-868. doi: 10.1063/1.532690. [41] J. Lenells, A variational approach to the stability of periodic peakons, J. Nonlinear Math. Phys., 11 (2004), 151-163. doi: 10.2991/jnmp.2004.11.2.2. [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), 27-63. doi: 10.1006/jdeq.1999.3683. [43] G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208. doi: 10.1016/S0393-0440(97)00010-7. [44] L. Tian, Y. Wang and J. Zhou, Global conservative and dissipative solutions of a coupled Camassa-Holm equations, J. Math. Phys., 52 (2011), 063702, 29 pp. doi: 10.1063/1.3600216. [45] L. Tian and Y. Xu, Attractor for a viscous coupled Camassa-Holm equation, Adv. Differ. Equ., (2010), 512812, 30 pp. [46] J. F. Toland, Stokes waves, Topol. Methods. Nonlinear Anal., 7 (1996), 1-48. [47] K. Yan and Z. Yin, Analytic solutions of the Cauchy problem for two-component shallow water systems, Math. Z., 269 (2011), 1113-1127. doi: 10.1007/s00209-010-0775-5. [48] M. Zhu and Blow-up, Global Existence and Persistence Properties for the Coupled Camassa-Holm equations, Math Phys. Anal Geom., 14 (2011), 197-209. doi: 10.1007/s11040-011-9094-2.

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##### References:
 [1] M. Baouendi and C. Goulaouic, Remarks on the abstract form of nonlinear Cauchy-Kowalevski theorems, Commun. Partial Differential Equation, 2 (1977), 1151-1162. doi: 10.1080/03605307708820057. [2] M. Baouendi and C. Goulaouic, Sharp estimates for analytic pseudodifferential operators and application to the Cauchy problems, J. Differential Equations, 48 (1983), 241-268. doi: 10.1016/0022-0396(83)90051-7. [3] 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. [4] A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27. doi: 10.1142/S0219530507000857. [5] R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661. [6] J. Chemin, Localization in Fourier space and Navier-Stokes system, Phase Space Analysis of Partial Differential Equations. Proceedings, CRM series, Pisa, (2004), 53-135. [7] A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362. doi: 10.5802/aif.1757. [8] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5. [9] A. Constantin, On the inverse spectral problem for the Camassa-Holm equation, J. Funct. Anal., 155 (1998), 352-363. doi: 10.1006/jfan.1997.3231. [10] A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431. doi: 10.1090/S0273-0979-07-01159-7. [11] A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12. [12] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586. [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), 75-91. doi: 10.1007/PL00004793. [14] A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa., 26 (1998), 303-328. [15] A. Constantin, V. Gerdjikov and R. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207. doi: 10.1088/0266-5611/22/6/017. [16] A. Constantin and R. Ivanov, On an integrable two-component Camassa-Holm shallow water system, Phys. Lett. A., 372 (2008), 7129-7132. doi: 10.1016/j.physleta.2008.10.050. [17] A. Constantin, T. Kappeler, B. Kolev and P. Topalov, On geodesic exponential maps of the Virasoro group, Ann. Global Anal. Geom., 31 (2007), 155-180. doi: 10.1007/s10455-006-9042-8. [18] A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Commentarii Mathematici Helvetici, 78 (2003), 787-804. doi: 10.1007/s00014-003-0785-6. [19] 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. [20] A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D. [21] A. Constantin and W. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. [22] R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988. [23] R. Danchin, Fourier Analysis Methods for PDEs, Lecture Notes, 14 November, 2003. [24] R. Dachin, A note on well-posedness for Camassa-Holm equation, J. Differential Equations, 192 (2003), 429-444. doi: 10.1016/S0022-0396(03)00096-2. [25] J. Escher and Z. Yin, Initial boundary value problems for nonlinear dispersive wave equations, J. Funct. Anal., 256 (2009), 479-508. doi: 10.1016/j.jfa.2008.07.010. [26] J. Escher and Z. Yin, Initial boundary value problems of the Camassa-Holm equation, Commun. Partial Differential Equation, 33 (2008), 377-395. doi: 10.1080/03605300701318872. [27] Y. Fu, G. Gui, Y. Liu and C. Qu, On the Cauchy problem for the integrable Camassa-Holm type equation with cubic nonlinearity, arXiv:1108.5368v2. [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), 415-448. doi: 10.1007/s00208-010-0483-9. [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), 012906, 1-25. doi: 10.1063/1.3064810. [30] Y. Fu and C. Qu, On a new Three-Component Camassa-Holm equation with peakons, Comm. Theor. Phys., 53 (2010), 223-230. [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), 4251-4278. doi: 10.1016/j.jfa.2010.02.008. [32] 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. [33] A. Himonas and G. Misiolek, Analyticity of the Cauchy problem for an integrable evolution equation, Math. Ann., 327 (2003), 575-584. doi: 10.1007/s00208-003-0466-1. [34] H. Holden and X. Raynaud, Global conservative solutions of the Camassa-Holm equations-a Lagrangianpoiny of view, Comm. Partial Differential Equations, 32 (2007), 1511-1549. doi: 10.1080/03605300601088674. [35] 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. [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), 451-467. doi: 10.1142/S0219891612500142. [37] D. Holm and R. Ivanov, Multi-component generalizations of the CH equation: geometrical aspects, peakons and numerical examples, J. Phys. A, 43 (2010), 492001, 1-21. doi: 10.1088/1751-8113/43/49/492001. [38] D. Holm and R. Ivanov, Two-component CH system: inverse scattering, peakons and geometry, Inverse Problems, 27 (2011), 045013, 1-21. doi: 10.1088/0266-5611/27/4/045013. [39] D. Holm, L. Onaraigh and C. Tronci, Singular solutions of a modified two-component Camassa-Holm equation, Phys. Rev. E., 79 (2009), 016601, 1-9. doi: 10.1103/PhysRevE.79.016601. [40] S. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group, J. Math. Phys., 40 (1999), 857-868. doi: 10.1063/1.532690. [41] J. Lenells, A variational approach to the stability of periodic peakons, J. Nonlinear Math. Phys., 11 (2004), 151-163. doi: 10.2991/jnmp.2004.11.2.2. [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), 27-63. doi: 10.1006/jdeq.1999.3683. [43] G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208. doi: 10.1016/S0393-0440(97)00010-7. [44] L. Tian, Y. Wang and J. Zhou, Global conservative and dissipative solutions of a coupled Camassa-Holm equations, J. Math. Phys., 52 (2011), 063702, 29 pp. doi: 10.1063/1.3600216. [45] L. Tian and Y. Xu, Attractor for a viscous coupled Camassa-Holm equation, Adv. Differ. Equ., (2010), 512812, 30 pp. [46] J. F. Toland, Stokes waves, Topol. Methods. Nonlinear Anal., 7 (1996), 1-48. [47] K. Yan and Z. Yin, Analytic solutions of the Cauchy problem for two-component shallow water systems, Math. Z., 269 (2011), 1113-1127. doi: 10.1007/s00209-010-0775-5. [48] M. Zhu and Blow-up, Global Existence and Persistence Properties for the Coupled Camassa-Holm equations, Math Phys. Anal Geom., 14 (2011), 197-209. doi: 10.1007/s11040-011-9094-2.
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