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On the Cauchy problem of a three-component Camassa--Holm equations
1. | Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China |
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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