American Institute of Mathematical Sciences

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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[11]

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

[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.  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), 75-91. doi: 10.1007/PL00004793.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[29]

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

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[33]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[11]

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

[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.  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), 75-91. doi: 10.1007/PL00004793.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[29]

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

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[33]

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

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