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May  2016, 36(5): 2827-2854. doi: 10.3934/dcds.2016.36.2827

On the Cauchy problem of a three-component Camassa--Holm equations

Xinglong Wu 1,

1. 

Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China

Received  March 2015 Revised  April 2015 Published  October 2015

The present paper is mainly concerned with the well-posedness, blow-up phenomena and exponential decay of solution. The well-posedness for a three-component Camassa--Holm equation is established in a critical Besov space. Comparing with the result of Hu, ect. in the paper [25], a new wave-breaking solution is obtained. The exponential decay of solution in our paper covers and extents the corresponding results in [12,24,31].
Keywords: A three-component Camassa--Holm equations, blow-up phenomena, the exponential decay, a critical Besov space, traveling wave solutions., wave-breaking.
Mathematics Subject Classification: 35G25, 35L0.
Citation: Xinglong Wu. On the Cauchy problem of a three-component Camassa--Holm equations. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2827-2854. doi: 10.3934/dcds.2016.36.2827
References:
[1]

H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer-Verlag Berlin Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

[2]

R. Beals, D. Sattinger and J. Szmigielski, Multipeakons and a theorem of Stieltjes, Inverse Problems, 15 (1999), L1-L4. doi: 10.1088/0266-5611/15/1/001.

[3]

L. Brandolese, Breakdown for the Camassa-Holm equation using decay criteria and persistence in weighted spaces, Int. Math. Res. Notices, 22 (2012), 5161-5181.

[4]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Rat. Mech. Anal., 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z.

[5]

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

[6]

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

[7]

R. Camassa, D. Holm and J. Hyman, An integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. doi: 10.1016/S0065-2156(08)70254-0.

[8]

A. Constantin, Global existence and breaking waves for a shallow water equation: A geometric approch, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362. doi: 10.5802/aif.1757.

[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 J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586.

[12]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa, 26 (1998), 303-328.

[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, Analyticity of periodic traveling free surface water waves with vorticity, Ann. Math., 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12.

[15]

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.

[16]

A. Constantin and W. A. 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.

[17]

H. H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mechanica, 127 (1998), 193-207. doi: 10.1007/BF01170373.

[18]

R. Danchin, A note on well-posedness for Camassa-Holm equation, J. Differential Equations, 192 (2003), 429-444. doi: 10.1016/S0022-0396(03)00096-2.

[19]

R. Danchin, Fourier Analysis Methods for PDEs, Lecture Notes, 14 November, 2005.

[20]

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, 25pp. doi: 10.1063/1.3064810.

[21]

B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa-Holm equation, Physica D, 95 (1996), 229-243. doi: 10.1016/0167-2789(96)00048-6.

[22]

A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries, Physica D, 4 (1981), 47-66. doi: 10.1016/0167-2789(81)90004-X.

[23]

A. E. Green and P. M. Naghdi, A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech., 78 (1976), 237-246. doi: 10.1017/S0022112076002425.

[24]

A. Himonas, G. Misiolek, G. Ponce and Y. Zhou, Persistence properties and unique continuation of solutions of the Camassa-Holm equation, Comm. Math. Phys., 271 (2007), 511-522. doi: 10.1007/s00220-006-0172-4.

[25]

Q. Y. Hu, L. Y. Lin and J. Jin, Well-posedness and blowup phenomena for a three-component Camassa-Holm system with peakons, J. Hyperbolic differential Equations, 9 (2012), 451-467. doi: 10.1142/S0219891612500142.

[26]

T. Kato, Quasi-linear equation of evolution, with applications to partical differential equations, in Spectral Theorey and Differential Equation, Lecture Notes in Math., 488, Spring-Verlag, Berlin, 1975, 25-70.

[27]

T. Kato, On the Korteweg-de Vries equation, Manuscripta Math., 28 (1979), 89-99. doi: 10.1007/BF01647967.

[28]

T. Kato, On the Cauchy problem for the generalized Korteweg-de Vries equation, in Studies in Applied Mathematics, Adv. Math. Suppl. Stu., 8, Academic Press, New York, 1983, 93-128.

[29]

B. G. Pachpatte, Inequalities for Differential and Integral Equations, Academic Press, Inc., San Diego, CA, 1998.

[30]

C. Qu and Y. Fu, On a Three-component Camassa-Holm equation with peakons, Commun. Theor. Phys., 53 (2010), 223-230.

[31]

X. Wu and B. Guo, Persistence properties and infinite propagation for the modified 2-component Camassa-Holm equation, Discrete Contin. Dyn. Syst. A, 33 (2013), 3211-3223. doi: 10.3934/dcds.2013.33.3211.

[32]

X. Wu and Z. Yin, A note on the Cauchy problem of the Novikov equation, Appl. Anal., 92 (2013), 1116-1137. doi: 10.1080/00036811.2011.649735.

[33]

X. Wu and Z. Yin, Well-posedness and global existence for the Novikov equation, Annali Sc. Norm. Sup. Pisa, 11 (2012), 707-727.

show all references

References:
[1]

H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer-Verlag Berlin Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

[2]

R. Beals, D. Sattinger and J. Szmigielski, Multipeakons and a theorem of Stieltjes, Inverse Problems, 15 (1999), L1-L4. doi: 10.1088/0266-5611/15/1/001.

[3]

L. Brandolese, Breakdown for the Camassa-Holm equation using decay criteria and persistence in weighted spaces, Int. Math. Res. Notices, 22 (2012), 5161-5181.

[4]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Rat. Mech. Anal., 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z.

[5]

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

[6]

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

[7]

R. Camassa, D. Holm and J. Hyman, An integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. doi: 10.1016/S0065-2156(08)70254-0.

[8]

A. Constantin, Global existence and breaking waves for a shallow water equation: A geometric approch, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362. doi: 10.5802/aif.1757.

[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 J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586.

[12]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa, 26 (1998), 303-328.

[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, Analyticity of periodic traveling free surface water waves with vorticity, Ann. Math., 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12.

[15]

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.

[16]

A. Constantin and W. A. 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.

[17]

H. H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mechanica, 127 (1998), 193-207. doi: 10.1007/BF01170373.

[18]

R. Danchin, A note on well-posedness for Camassa-Holm equation, J. Differential Equations, 192 (2003), 429-444. doi: 10.1016/S0022-0396(03)00096-2.

[19]

R. Danchin, Fourier Analysis Methods for PDEs, Lecture Notes, 14 November, 2005.

[20]

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, 25pp. doi: 10.1063/1.3064810.

[21]

B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa-Holm equation, Physica D, 95 (1996), 229-243. doi: 10.1016/0167-2789(96)00048-6.

[22]

A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries, Physica D, 4 (1981), 47-66. doi: 10.1016/0167-2789(81)90004-X.

[23]

A. E. Green and P. M. Naghdi, A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech., 78 (1976), 237-246. doi: 10.1017/S0022112076002425.

[24]

A. Himonas, G. Misiolek, G. Ponce and Y. Zhou, Persistence properties and unique continuation of solutions of the Camassa-Holm equation, Comm. Math. Phys., 271 (2007), 511-522. doi: 10.1007/s00220-006-0172-4.

[25]

Q. Y. Hu, L. Y. Lin and J. Jin, Well-posedness and blowup phenomena for a three-component Camassa-Holm system with peakons, J. Hyperbolic differential Equations, 9 (2012), 451-467. doi: 10.1142/S0219891612500142.

[26]

T. Kato, Quasi-linear equation of evolution, with applications to partical differential equations, in Spectral Theorey and Differential Equation, Lecture Notes in Math., 488, Spring-Verlag, Berlin, 1975, 25-70.

[27]

T. Kato, On the Korteweg-de Vries equation, Manuscripta Math., 28 (1979), 89-99. doi: 10.1007/BF01647967.

[28]

T. Kato, On the Cauchy problem for the generalized Korteweg-de Vries equation, in Studies in Applied Mathematics, Adv. Math. Suppl. Stu., 8, Academic Press, New York, 1983, 93-128.

[29]

B. G. Pachpatte, Inequalities for Differential and Integral Equations, Academic Press, Inc., San Diego, CA, 1998.

[30]

C. Qu and Y. Fu, On a Three-component Camassa-Holm equation with peakons, Commun. Theor. Phys., 53 (2010), 223-230.

[31]

X. Wu and B. Guo, Persistence properties and infinite propagation for the modified 2-component Camassa-Holm equation, Discrete Contin. Dyn. Syst. A, 33 (2013), 3211-3223. doi: 10.3934/dcds.2013.33.3211.

[32]

X. Wu and Z. Yin, A note on the Cauchy problem of the Novikov equation, Appl. Anal., 92 (2013), 1116-1137. doi: 10.1080/00036811.2011.649735.

[33]

X. Wu and Z. Yin, Well-posedness and global existence for the Novikov equation, Annali Sc. Norm. Sup. Pisa, 11 (2012), 707-727.

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