November  2011, 30(4): 1161-1180. doi: 10.3934/dcds.2011.30.1161

Some remarks for a modified periodic Camassa-Holm system

1. 

Department of Mathematics, Southeast University, Nanjing 210018

2. 

Natural Science Research Center, Harbin Institute of Technology, Harbin 150080, China

Received  May 2010 Revised  August 2010 Published  May 2011

This paper is concerned with a modified two-component periodic Camassa-Holm system. The local well-posedness and low regularity result of solution are established by using the techniques of pseudoparabolic regularization and some priori estimates derived from the equation itself. A wave-breaking for strong solutions and several results of blow-up solution with certain initial profiles are described. In addition, the initial boundary value problem for a modified two-component periodic Camassa-Holm system is also considered.
Citation: Guangying Lv, Mingxin Wang. Some remarks for a modified periodic Camassa-Holm system. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1161-1180. doi: 10.3934/dcds.2011.30.1161
References:
[1]

J. Bona and R. Smith, The initial value problem for the Korteweg-de Vries equation,, Phil. Trans. Roy. Soc. London A, 278 (1975), 555.  doi: 10.1098/rsta.1975.0035.  Google Scholar

[2]

A. Bressan and A. Constantin, Global conservation solutions of the Camassa-Holm equation,, Arch. Rat. 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,, Anal. Appl., 5 (2007), 1.  doi: 10.1142/S0219530507000857.  Google Scholar

[4]

R. Cammasa 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

[5]

M. Chen, S. Liu and Y. Zhang, A 2-component generalization of the Camassa-Holm equation and its solutions,, Lett. Math. Phys., 75 (2006), 1.  doi: 10.1007/s11005-005-0041-7.  Google Scholar

[6]

A. Constantin, The hamiltonian structure of the Camassa-Holm equation,, Expo. Math., 15 (1997), 53.   Google Scholar

[7]

A. Constantin, On the inverse spectral problem for the Camasa-Holm equation,, J. Funct. Anal., 155 (1998), 352.  doi: 10.1006/jfan.1997.3231.  Google Scholar

[8]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach,, Ann. Inst. Fourier., 50 (2000), 321.   Google Scholar

[9]

A. Constantin, On the scattering problem for the Camassa-Holm equation,, Proc. R. Soc. London A, 457 (2001), 953.  doi: 10.1098/rspa.2000.0701.  Google Scholar

[10]

A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523.  doi: 10.1007/s00222-006-0002-5.  Google Scholar

[11]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Rat. Mech. Anal., 1992 (2009), 165.  doi: 10.1007/s00205-008-0128-2.  Google Scholar

[12]

A. Constantin and J. Escher, Well-posedness, global existence and blowup phenomena for a periodic quasi-linear hyperbolic equation,, Comm. Pure Appl. Math., 51 (1998), 475.  doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5.  Google Scholar

[13]

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

[14]

A. Constantin and J. Escher, Global weak solutions for a shallow water equation,, Indiana Univ. Math. J., 47 (1998), 1527.  doi: 10.1512/iumj.1998.47.1466.  Google Scholar

[15]

A. Constantin and J. Escher, On the blow-up rate and the blow-up of breaking waves for a shallow water equation,, Math. Z., 233 (2000), 75.  doi: 10.1007/PL00004793.  Google Scholar

[16]

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

[17]

A. Constantin, V. Gerdjikov and R. I. Ivanov, Inverse scattering transform for the Camassa-Holm equation,, Inverse Problems, 22 (2006), 2197.  doi: 10.1088/0266-5611/22/6/017.  Google Scholar

[18]

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

[19]

A. Constantin and R. S. Johnson, Propagation of very long water waves, with vorticity over variable depth, with applicastions to tsuami,, Fluid Dynam. Res., 40 (2008), 175.  doi: 10.1016/j.fluiddyn.2007.06.004.  Google Scholar

[20]

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

[21]

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

[22]

A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle,, Comment. Math. Helv., 78 (2003), 787.  doi: 10.1007/s00014-003-0785-6.  Google Scholar

[23]

A. Constantin and H. P. McKean, A shallow water equation on the circle,, Comm. Pure Appl. Math., 52 (1998), 949.  doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D.  Google Scholar

[24]

A. Constantin and L. Molinet, Global weak solutions for a shallow water equation,, Comm. Math. Phys., 211 (2000), 45.  doi: 10.1007/s002200050801.  Google Scholar

[25]

G. Coclite, H. Holden and K. Karlsen, Global weak solutions to a generalized hyperelastic-rod wave equation,, SIAM J. Math. Anal., 37 (2005), 1044.  doi: 10.1137/040616711.  Google Scholar

[26]

R. Danchin, A few remarks on the Camassa-Holm equation,, Differential Integral Equations, 14 (2001), 953.   Google Scholar

[27]

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.   Google Scholar

[28]

J. Escher and Z. Yin, Initial boundary value problems of the Camassa-Holm equation,, Commun. Partial Differential Equation, 33 (2008), 377.   Google Scholar

[29]

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

[30]

Y. Fu and C. Qu, Well-posedness and blow-up solution for a new coupled Camassa-Holm system with peakons,, J. Math. Phys., 50 (2009).   Google Scholar

[31]

Y. Fu, Y. Liu and C. Qu, Well-posedness and blow-up solution for a modified two-component periodic Camassa-Holm system with peakons,, Math. Ann., 348 (2010), 415.  doi: 10.1007/s00208-010-0483-9.  Google Scholar

[32]

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.   Google Scholar

[33]

C. Guan, K. Karlsen and Z. Yin, Well-posedness and blow-up phenomena for a modified two-component Camassa-Holm equation,, Contemporary Mathematics, 526 (2010).   Google Scholar

[34]

G. Gui and Y. Liu, On the cauchy problem for the two-component Camassa-Holm system,, Math. Z., ().  doi: 10.1007/s00209-009-0660-2.  Google Scholar

[35]

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

[36]

D. Henry, Infinite 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

[37]

H. Holden and X. Raynaud, Global conservation solutions of the Camassa-Holm equation-A lagrangian point of view,, Commun. Partial Differential Equation, 32 (2007), 1511.   Google Scholar

[38]

D. Holm, L. Naraigh and C. Tronci, Singular solution of a modified two component Camassa-Holm equation,, Phy. Rev. E, 79 (2009), 1.  doi: 10.1103/PhysRevE.79.016601.  Google Scholar

[39]

D. Ionescu-Kruse, Variational derivation of the Camassa-Holm shallow water equation,, J. Nonlinear Math. Phys., 14 (2007), 303.  doi: 10.2991/jnmp.2007.14.3.1.  Google Scholar

[40]

R. Iorio and de Magãlhaes Iorio, "Fourier Analysis and Partial Differential Equation,", Cambridge University Press, (2001).   Google Scholar

[41]

R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid Mech., 457 (2002), 63.   Google Scholar

[42]

T. Kato, Quasi-linear equation of evolution, with application to partial differential equation,, in, 448 (1975), 25.   Google Scholar

[43]

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

[44]

M. Lakshmanan, Integrable nonlinear wave equations and possible connections to tsunami dynamics,, Tsunami and Nonlinear Waves, (2007), 31.  doi: 10.1007/978-3-540-71256-5_2.  Google Scholar

[45]

Y. Li and P. Oliver, Well-posedness and blow-up solutions for an integrable nonlinear dispersive model wave equation,, J. Differential Equations, 162 (2000), 27.   Google Scholar

[46]

J. L. Lions, "Quelques Méthodes de Résolution des Problèaux Limites Nonlinéaires,", Dunod, (1969).   Google Scholar

[47]

H. P. McKean, Integrable systems and algebraic curves,, Lecture Notes in Math., 755 (1979), 83.   Google Scholar

[48]

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

[49]

V. Ovsienko and B. Khesin, The super Korteweg-de Vries equation as an Euler equation,, Funktsional. Anal. i Prilozhen, 21 (1987), 81.   Google Scholar

[50]

H. Segur, Waves in shallow water, with emphasis int the tsunami of 2004,, Tsunami and Nonlinear Waves, (2007), 31.   Google Scholar

[51]

W. Walter, "Differential and Integral Inequalities,", "Differential and Integral Inequalities,", (1970).   Google Scholar

[52]

Z. Xin and P. Zhang, On the weak solutions to a shallow water equation,, Comm. Pure Appl. Math., 53 (2000), 1411.  doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5.  Google Scholar

[53]

Z. Yin, Well-posedness, global existence phenomena for an integrable shallow water equation,, Discrete Contin. Dyn. Syst., 10 (2004), 393.  doi: 10.3934/dcds.2004.11.393.  Google Scholar

[54]

Z. Yin, Global existence for a new periodic integrable equation,, J. Math. Anal. Appl., 283 (2003), 129.  doi: 10.1016/S0022-247X(03)00250-6.  Google Scholar

show all references

References:
[1]

J. Bona and R. Smith, The initial value problem for the Korteweg-de Vries equation,, Phil. Trans. Roy. Soc. London A, 278 (1975), 555.  doi: 10.1098/rsta.1975.0035.  Google Scholar

[2]

A. Bressan and A. Constantin, Global conservation solutions of the Camassa-Holm equation,, Arch. Rat. 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,, Anal. Appl., 5 (2007), 1.  doi: 10.1142/S0219530507000857.  Google Scholar

[4]

R. Cammasa 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

[5]

M. Chen, S. Liu and Y. Zhang, A 2-component generalization of the Camassa-Holm equation and its solutions,, Lett. Math. Phys., 75 (2006), 1.  doi: 10.1007/s11005-005-0041-7.  Google Scholar

[6]

A. Constantin, The hamiltonian structure of the Camassa-Holm equation,, Expo. Math., 15 (1997), 53.   Google Scholar

[7]

A. Constantin, On the inverse spectral problem for the Camasa-Holm equation,, J. Funct. Anal., 155 (1998), 352.  doi: 10.1006/jfan.1997.3231.  Google Scholar

[8]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach,, Ann. Inst. Fourier., 50 (2000), 321.   Google Scholar

[9]

A. Constantin, On the scattering problem for the Camassa-Holm equation,, Proc. R. Soc. London A, 457 (2001), 953.  doi: 10.1098/rspa.2000.0701.  Google Scholar

[10]

A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523.  doi: 10.1007/s00222-006-0002-5.  Google Scholar

[11]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Rat. Mech. Anal., 1992 (2009), 165.  doi: 10.1007/s00205-008-0128-2.  Google Scholar

[12]

A. Constantin and J. Escher, Well-posedness, global existence and blowup phenomena for a periodic quasi-linear hyperbolic equation,, Comm. Pure Appl. Math., 51 (1998), 475.  doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5.  Google Scholar

[13]

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

[14]

A. Constantin and J. Escher, Global weak solutions for a shallow water equation,, Indiana Univ. Math. J., 47 (1998), 1527.  doi: 10.1512/iumj.1998.47.1466.  Google Scholar

[15]

A. Constantin and J. Escher, On the blow-up rate and the blow-up of breaking waves for a shallow water equation,, Math. Z., 233 (2000), 75.  doi: 10.1007/PL00004793.  Google Scholar

[16]

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

[17]

A. Constantin, V. Gerdjikov and R. I. Ivanov, Inverse scattering transform for the Camassa-Holm equation,, Inverse Problems, 22 (2006), 2197.  doi: 10.1088/0266-5611/22/6/017.  Google Scholar

[18]

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

[19]

A. Constantin and R. S. Johnson, Propagation of very long water waves, with vorticity over variable depth, with applicastions to tsuami,, Fluid Dynam. Res., 40 (2008), 175.  doi: 10.1016/j.fluiddyn.2007.06.004.  Google Scholar

[20]

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

[21]

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

[22]

A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle,, Comment. Math. Helv., 78 (2003), 787.  doi: 10.1007/s00014-003-0785-6.  Google Scholar

[23]

A. Constantin and H. P. McKean, A shallow water equation on the circle,, Comm. Pure Appl. Math., 52 (1998), 949.  doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D.  Google Scholar

[24]

A. Constantin and L. Molinet, Global weak solutions for a shallow water equation,, Comm. Math. Phys., 211 (2000), 45.  doi: 10.1007/s002200050801.  Google Scholar

[25]

G. Coclite, H. Holden and K. Karlsen, Global weak solutions to a generalized hyperelastic-rod wave equation,, SIAM J. Math. Anal., 37 (2005), 1044.  doi: 10.1137/040616711.  Google Scholar

[26]

R. Danchin, A few remarks on the Camassa-Holm equation,, Differential Integral Equations, 14 (2001), 953.   Google Scholar

[27]

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.   Google Scholar

[28]

J. Escher and Z. Yin, Initial boundary value problems of the Camassa-Holm equation,, Commun. Partial Differential Equation, 33 (2008), 377.   Google Scholar

[29]

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

[30]

Y. Fu and C. Qu, Well-posedness and blow-up solution for a new coupled Camassa-Holm system with peakons,, J. Math. Phys., 50 (2009).   Google Scholar

[31]

Y. Fu, Y. Liu and C. Qu, Well-posedness and blow-up solution for a modified two-component periodic Camassa-Holm system with peakons,, Math. Ann., 348 (2010), 415.  doi: 10.1007/s00208-010-0483-9.  Google Scholar

[32]

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.   Google Scholar

[33]

C. Guan, K. Karlsen and Z. Yin, Well-posedness and blow-up phenomena for a modified two-component Camassa-Holm equation,, Contemporary Mathematics, 526 (2010).   Google Scholar

[34]

G. Gui and Y. Liu, On the cauchy problem for the two-component Camassa-Holm system,, Math. Z., ().  doi: 10.1007/s00209-009-0660-2.  Google Scholar

[35]

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

[36]

D. Henry, Infinite 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

[37]

H. Holden and X. Raynaud, Global conservation solutions of the Camassa-Holm equation-A lagrangian point of view,, Commun. Partial Differential Equation, 32 (2007), 1511.   Google Scholar

[38]

D. Holm, L. Naraigh and C. Tronci, Singular solution of a modified two component Camassa-Holm equation,, Phy. Rev. E, 79 (2009), 1.  doi: 10.1103/PhysRevE.79.016601.  Google Scholar

[39]

D. Ionescu-Kruse, Variational derivation of the Camassa-Holm shallow water equation,, J. Nonlinear Math. Phys., 14 (2007), 303.  doi: 10.2991/jnmp.2007.14.3.1.  Google Scholar

[40]

R. Iorio and de Magãlhaes Iorio, "Fourier Analysis and Partial Differential Equation,", Cambridge University Press, (2001).   Google Scholar

[41]

R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid Mech., 457 (2002), 63.   Google Scholar

[42]

T. Kato, Quasi-linear equation of evolution, with application to partial differential equation,, in, 448 (1975), 25.   Google Scholar

[43]

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

[44]

M. Lakshmanan, Integrable nonlinear wave equations and possible connections to tsunami dynamics,, Tsunami and Nonlinear Waves, (2007), 31.  doi: 10.1007/978-3-540-71256-5_2.  Google Scholar

[45]

Y. Li and P. Oliver, Well-posedness and blow-up solutions for an integrable nonlinear dispersive model wave equation,, J. Differential Equations, 162 (2000), 27.   Google Scholar

[46]

J. L. Lions, "Quelques Méthodes de Résolution des Problèaux Limites Nonlinéaires,", Dunod, (1969).   Google Scholar

[47]

H. P. McKean, Integrable systems and algebraic curves,, Lecture Notes in Math., 755 (1979), 83.   Google Scholar

[48]

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

[49]

V. Ovsienko and B. Khesin, The super Korteweg-de Vries equation as an Euler equation,, Funktsional. Anal. i Prilozhen, 21 (1987), 81.   Google Scholar

[50]

H. Segur, Waves in shallow water, with emphasis int the tsunami of 2004,, Tsunami and Nonlinear Waves, (2007), 31.   Google Scholar

[51]

W. Walter, "Differential and Integral Inequalities,", "Differential and Integral Inequalities,", (1970).   Google Scholar

[52]

Z. Xin and P. Zhang, On the weak solutions to a shallow water equation,, Comm. Pure Appl. Math., 53 (2000), 1411.  doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5.  Google Scholar

[53]

Z. Yin, Well-posedness, global existence phenomena for an integrable shallow water equation,, Discrete Contin. Dyn. Syst., 10 (2004), 393.  doi: 10.3934/dcds.2004.11.393.  Google Scholar

[54]

Z. Yin, Global existence for a new periodic integrable equation,, J. Math. Anal. Appl., 283 (2003), 129.  doi: 10.1016/S0022-247X(03)00250-6.  Google Scholar

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