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 and Continuous Dynamical Systems, 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), 555-601. doi: 10.1098/rsta.1975.0035.

[2]

A. Bressan and A. Constantin, Global conservation solutions of the Camassa-Holm equation, Arch. Rat. 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, Anal. Appl., 5 (2007), 1-27. doi: 10.1142/S0219530507000857.

[4]

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

[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), 1-15. doi: 10.1007/s11005-005-0041-7.

[6]

A. Constantin, The hamiltonian structure of the Camassa-Holm equation, Expo. Math., 15 (1997), 53-85.

[7]

A. Constantin, On the inverse spectral problem for the Camasa-Holm equation, J. Funct. Anal., 155 (1998), 352-363. doi: 10.1006/jfan.1997.3231.

[8]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier., 50 (2000), 321-362.

[9]

A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. R. Soc. London A, 457 (2001), 953-970. doi: 10.1098/rspa.2000.0701.

[10]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.

[11]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Rat. Mech. Anal., 1992 (2009), 165-186. doi: 10.1007/s00205-008-0128-2.

[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), 475-504. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5.

[13]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., (1998), 229-243. doi: 10.1007/BF02392586.

[14]

A. Constantin and J. Escher, Global weak solutions for a shallow water equation, Indiana Univ. Math. J., 47 (1998), 1527-1245. doi: 10.1512/iumj.1998.47.1466.

[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), 75-91. doi: 10.1007/PL00004793.

[16]

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.

[17]

A. Constantin, V. Gerdjikov and R. I. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207. doi: 10.1088/0266-5611/22/6/017.

[18]

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.

[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), 175-211. doi: 10.1016/j.fluiddyn.2007.06.004.

[20]

A. Constantin, T. Kappeler, B. Kolev and P. Topalov, On geodesic exponential maps of the Virasoro group, Ann. Global Anal. Geom., 31 (2007), 155-180. doi: 10.1007/s10455-006-9042-8.

[21]

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

[22]

A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804. doi: 10.1007/s00014-003-0785-6.

[23]

A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1998), 949-982. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D.

[24]

A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61. doi: 10.1007/s002200050801.

[25]

G. Coclite, H. Holden and K. Karlsen, Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM J. Math. Anal., 37 (2005), 1044-1069. doi: 10.1137/040616711.

[26]

R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988.

[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), 493-513.

[28]

J. Escher and Z. Yin, Initial boundary value problems of the Camassa-Holm equation, Commun. Partial Differential Equation, 33 (2008), 377-395.

[29]

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.

[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).

[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), 415-448. doi: 10.1007/s00208-010-0483-9.

[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), 2003-2014.

[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).

[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.

[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), 4251-4278. doi: 10.1016/j.jfa.2010.02.008.

[36]

D. Henry, Infinite 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.

[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), 1511-1549.

[38]

D. Holm, L. Naraigh and C. Tronci, Singular solution of a modified two component Camassa-Holm equation, Phy. Rev. E, 79 (2009), 1-13. doi: 10.1103/PhysRevE.79.016601.

[39]

D. Ionescu-Kruse, Variational derivation of the Camassa-Holm shallow water equation, J. Nonlinear Math. Phys., 14 (2007), 303-312. doi: 10.2991/jnmp.2007.14.3.1.

[40]

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

[41]

R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 457 (2002), 63-82.

[42]

T. Kato, Quasi-linear equation of evolution, with application to partial differential equation, in "Spectral Theory and Differential Equations," Lecture Notes in Math., 448, Springer Verlag, Berlin, (1975), 25-70.

[43]

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.

[44]

M. Lakshmanan, Integrable nonlinear wave equations and possible connections to tsunami dynamics, in "Tsunami and Nonlinear Waves" (Ed. A. Kundu), Springer, Berlin, (2007), 31-49. doi: 10.1007/978-3-540-71256-5_2.

[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), 27-63.

[46]

J. L. Lions, "Quelques Méthodes de Résolution des Problèaux Limites Nonlinéaires," Dunod, Gauthier-Villars, Paris, 1969.

[47]

H. P. McKean, Integrable systems and algebraic curves, Lecture Notes in Math., 755, Springer, Berlin, (1979), 83-200.

[48]

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.

[49]

V. Ovsienko and B. Khesin, The super Korteweg-de Vries equation as an Euler equation, Funktsional. Anal. i Prilozhen, 21 (1987), 81-82.

[50]

H. Segur, Waves in shallow water, with emphasis int the tsunami of 2004, in "Tsunami and Nonlinear Waves" (ed. A. Kundu), Springer, Berlin, (2007), 31-49.

[51]

W. Walter, "Differential and Integral Inequalities," Springer, New York, 1970.

[52]

Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433. doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5.

[53]

Z. Yin, Well-posedness, global existence phenomena for an integrable shallow water equation, Discrete Contin. Dyn. Syst., 10 (2004), 393-411. doi: 10.3934/dcds.2004.11.393.

[54]

Z. Yin, Global existence for a new periodic integrable equation, J. Math. Anal. Appl., 283 (2003), 129-139. doi: 10.1016/S0022-247X(03)00250-6.

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), 555-601. doi: 10.1098/rsta.1975.0035.

[2]

A. Bressan and A. Constantin, Global conservation solutions of the Camassa-Holm equation, Arch. Rat. 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, Anal. Appl., 5 (2007), 1-27. doi: 10.1142/S0219530507000857.

[4]

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

[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), 1-15. doi: 10.1007/s11005-005-0041-7.

[6]

A. Constantin, The hamiltonian structure of the Camassa-Holm equation, Expo. Math., 15 (1997), 53-85.

[7]

A. Constantin, On the inverse spectral problem for the Camasa-Holm equation, J. Funct. Anal., 155 (1998), 352-363. doi: 10.1006/jfan.1997.3231.

[8]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier., 50 (2000), 321-362.

[9]

A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. R. Soc. London A, 457 (2001), 953-970. doi: 10.1098/rspa.2000.0701.

[10]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.

[11]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Rat. Mech. Anal., 1992 (2009), 165-186. doi: 10.1007/s00205-008-0128-2.

[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), 475-504. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5.

[13]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., (1998), 229-243. doi: 10.1007/BF02392586.

[14]

A. Constantin and J. Escher, Global weak solutions for a shallow water equation, Indiana Univ. Math. J., 47 (1998), 1527-1245. doi: 10.1512/iumj.1998.47.1466.

[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), 75-91. doi: 10.1007/PL00004793.

[16]

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.

[17]

A. Constantin, V. Gerdjikov and R. I. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207. doi: 10.1088/0266-5611/22/6/017.

[18]

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.

[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), 175-211. doi: 10.1016/j.fluiddyn.2007.06.004.

[20]

A. Constantin, T. Kappeler, B. Kolev and P. Topalov, On geodesic exponential maps of the Virasoro group, Ann. Global Anal. Geom., 31 (2007), 155-180. doi: 10.1007/s10455-006-9042-8.

[21]

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

[22]

A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804. doi: 10.1007/s00014-003-0785-6.

[23]

A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1998), 949-982. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D.

[24]

A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61. doi: 10.1007/s002200050801.

[25]

G. Coclite, H. Holden and K. Karlsen, Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM J. Math. Anal., 37 (2005), 1044-1069. doi: 10.1137/040616711.

[26]

R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988.

[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), 493-513.

[28]

J. Escher and Z. Yin, Initial boundary value problems of the Camassa-Holm equation, Commun. Partial Differential Equation, 33 (2008), 377-395.

[29]

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.

[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).

[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), 415-448. doi: 10.1007/s00208-010-0483-9.

[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), 2003-2014.

[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).

[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.

[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), 4251-4278. doi: 10.1016/j.jfa.2010.02.008.

[36]

D. Henry, Infinite 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.

[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), 1511-1549.

[38]

D. Holm, L. Naraigh and C. Tronci, Singular solution of a modified two component Camassa-Holm equation, Phy. Rev. E, 79 (2009), 1-13. doi: 10.1103/PhysRevE.79.016601.

[39]

D. Ionescu-Kruse, Variational derivation of the Camassa-Holm shallow water equation, J. Nonlinear Math. Phys., 14 (2007), 303-312. doi: 10.2991/jnmp.2007.14.3.1.

[40]

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

[41]

R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 457 (2002), 63-82.

[42]

T. Kato, Quasi-linear equation of evolution, with application to partial differential equation, in "Spectral Theory and Differential Equations," Lecture Notes in Math., 448, Springer Verlag, Berlin, (1975), 25-70.

[43]

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.

[44]

M. Lakshmanan, Integrable nonlinear wave equations and possible connections to tsunami dynamics, in "Tsunami and Nonlinear Waves" (Ed. A. Kundu), Springer, Berlin, (2007), 31-49. doi: 10.1007/978-3-540-71256-5_2.

[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), 27-63.

[46]

J. L. Lions, "Quelques Méthodes de Résolution des Problèaux Limites Nonlinéaires," Dunod, Gauthier-Villars, Paris, 1969.

[47]

H. P. McKean, Integrable systems and algebraic curves, Lecture Notes in Math., 755, Springer, Berlin, (1979), 83-200.

[48]

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.

[49]

V. Ovsienko and B. Khesin, The super Korteweg-de Vries equation as an Euler equation, Funktsional. Anal. i Prilozhen, 21 (1987), 81-82.

[50]

H. Segur, Waves in shallow water, with emphasis int the tsunami of 2004, in "Tsunami and Nonlinear Waves" (ed. A. Kundu), Springer, Berlin, (2007), 31-49.

[51]

W. Walter, "Differential and Integral Inequalities," Springer, New York, 1970.

[52]

Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433. doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5.

[53]

Z. Yin, Well-posedness, global existence phenomena for an integrable shallow water equation, Discrete Contin. Dyn. Syst., 10 (2004), 393-411. doi: 10.3934/dcds.2004.11.393.

[54]

Z. Yin, Global existence for a new periodic integrable equation, J. Math. Anal. Appl., 283 (2003), 129-139. doi: 10.1016/S0022-247X(03)00250-6.

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