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Local well-posedness in the critical Besov space and persistence properties for a three-component Camassa-Holm system with N-peakon solutions
1. | Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China |
2. | Department of Mathematics, Zhongshan University, Guangzhou, 510275 |
References:
[1] |
H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, Vol. 343, Berlin-Heidelberg-NewYork: Springer, 2011.
doi: 10.1007/978-3-642-16830-7. |
[2] |
L. Brandolese, Break down for the Camassa-Holm equation using decay criteria and persistence in weighted spaces, Int. Math. Res. Not. IMRN, 22 (2012), 5161-5181. |
[3] |
A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239.
doi: 10.1007/s00205-006-0010-z. |
[4] |
A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27.
doi: 10.1142/S0219530507000857. |
[5] |
R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
[6] |
R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33.
doi: 10.1016/S0065-2156(08)70254-0. |
[7] |
A. Constantin, The Hamiltonian structure of the Camassa-Holm equation, Exposition. Math., 15 (1997), 53-85. |
[8] |
A. Constantin, On the scattering problem for the Camassa-Holm equation, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 953-970.
doi: 10.1098/rspa.2000.0701. |
[9] |
A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.
doi: 10.5802/aif.1757. |
[10] |
A. Constantin, Finite propagation speed for the Camassa-Holm equation, J. Math. Phys., 46 (2005), 023506, 4 pp.
doi: 10.1063/1.1845603. |
[11] |
A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.
doi: 10.1007/s00222-006-0002-5. |
[12] |
A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 303-328. |
[13] |
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. |
[14] |
A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.
doi: 10.1007/BF02392586. |
[15] |
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. |
[16] |
A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568.
doi: 10.4007/annals.2011.173.1.12. |
[17] |
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. |
[18] |
A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61.
doi: 10.1007/s002200050801. |
[19] |
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. |
[20] |
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. |
[21] |
R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988. |
[22] |
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. |
[23] |
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.
doi: 10.3934/dcds.2007.19.493. |
[24] |
B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries,, Phys. D, 4 (): 47.
doi: 10.1016/0167-2789(81)90004-X. |
[25] |
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.
doi: 10.1016/j.jde.2009.08.002. |
[26] |
C. Guan, K. H. Karlsen and Z. Yin, Well-posedness and blow-up phenomena for a modified two-component Camassa-Holm equation, Contemp. Math., 526 (2010), 199-220.
doi: 10.1090/conm/526/10382. |
[27] |
C. Guan and Z. Yin, Global weak solutions for a two-component Camassa- Holm shallow water system, J. Funct. Anal., 260 (2011), 1132-1154.
doi: 10.1016/j.jfa.2010.11.015. |
[28] |
C. Guan and Z. Yin, Global weak solutions for a modified two-component Camassa-Holm equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 623-641.
doi: 10.1016/j.anihpc.2011.04.003. |
[29] |
C. Guan, H. He and Z. Yin, Well-posedness, blow-up phenomena and persistence properties for a two-component water wave system, Nonlinear Anal. Real World Appl., 25 (2015), 219-237.
doi: 10.1016/j.nonrwa.2015.04.001. |
[30] |
X. G. Geng and B. Xue, A three-component generalization of Camassa-Holm equation with N-peakon solutions, Adv. Math., 226 (2011), 827-839.
doi: 10.1016/j.aim.2010.07.009. |
[31] |
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. |
[32] |
D. Henry, Persistence properties for a family of nonlinear partial differential equations, Nonlinear Anal., 70 (2009), 1565-1573.
doi: 10.1016/j.na.2008.02.104. |
[33] |
D. D. Holm, L. Naraigh and C. Tronci, Singular solution of a modified twocomponent Camassa-Holm equation, Phys. Rev. E (3), 79 (2009), 016601, 13 pp.
doi: 10.1103/PhysRevE.79.016601. |
[34] |
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. |
[35] |
Y. Liu and Z. Yin, Global existence and blow-up phenomena for the Degasperis-Procesi equation, Comm. Math. Phys., 267 (2006), 801-820.
doi: 10.1007/s00220-006-0082-5. |
[36] |
W. Luo and Z. Yin, Global existence and local well-posedness for a three-component Camassa-Holm system with N-peakon solutions, J. Differential Equations, 259 (2015), 201-234.
doi: 10.1016/j.jde.2015.02.005. |
[37] |
G. Rodríguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46 (2001), 309-327.
doi: 10.1016/S0362-546X(01)00791-X. |
[38] |
W. Tan and Z. Yin, Global conservative solutions of a modified two-component Camassa-Holm shallow water system, J. Differential Equations, 251 (2011), 3558-3582.
doi: 10.1016/j.jde.2011.08.010. |
[39] |
W. Tan and Z. Yin, Global dissipative solutions of a modified two-component Camassa-Holm shallow water system, J. Math. Phys., 52 (2011), 033507, 24pp.
doi: 10.1063/1.3562928. |
[40] |
J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. |
[41] |
X. Wu and B. Guo, Persistence properties and infinite propagation for the modified 2-component Camassa-Holm equation, Discrete Contin. Dyn. Syst., 33 (2013), 3211-3223.
doi: 10.3934/dcds.2013.33.3211. |
[42] |
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. |
[43] |
K. Yan and Z. Yin, Well-posedness for a modified two-component Camassa-Holm system in critical spaces, Discrete Contin. Dyn. Syst., 33 (2013), 1699-1712.
doi: 10.3934/dcds.2013.33.1699. |
show all references
References:
[1] |
H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, Vol. 343, Berlin-Heidelberg-NewYork: Springer, 2011.
doi: 10.1007/978-3-642-16830-7. |
[2] |
L. Brandolese, Break down for the Camassa-Holm equation using decay criteria and persistence in weighted spaces, Int. Math. Res. Not. IMRN, 22 (2012), 5161-5181. |
[3] |
A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239.
doi: 10.1007/s00205-006-0010-z. |
[4] |
A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27.
doi: 10.1142/S0219530507000857. |
[5] |
R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
[6] |
R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33.
doi: 10.1016/S0065-2156(08)70254-0. |
[7] |
A. Constantin, The Hamiltonian structure of the Camassa-Holm equation, Exposition. Math., 15 (1997), 53-85. |
[8] |
A. Constantin, On the scattering problem for the Camassa-Holm equation, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 953-970.
doi: 10.1098/rspa.2000.0701. |
[9] |
A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.
doi: 10.5802/aif.1757. |
[10] |
A. Constantin, Finite propagation speed for the Camassa-Holm equation, J. Math. Phys., 46 (2005), 023506, 4 pp.
doi: 10.1063/1.1845603. |
[11] |
A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.
doi: 10.1007/s00222-006-0002-5. |
[12] |
A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 303-328. |
[13] |
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. |
[14] |
A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.
doi: 10.1007/BF02392586. |
[15] |
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. |
[16] |
A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568.
doi: 10.4007/annals.2011.173.1.12. |
[17] |
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. |
[18] |
A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61.
doi: 10.1007/s002200050801. |
[19] |
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. |
[20] |
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. |
[21] |
R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988. |
[22] |
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. |
[23] |
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.
doi: 10.3934/dcds.2007.19.493. |
[24] |
B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries,, Phys. D, 4 (): 47.
doi: 10.1016/0167-2789(81)90004-X. |
[25] |
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.
doi: 10.1016/j.jde.2009.08.002. |
[26] |
C. Guan, K. H. Karlsen and Z. Yin, Well-posedness and blow-up phenomena for a modified two-component Camassa-Holm equation, Contemp. Math., 526 (2010), 199-220.
doi: 10.1090/conm/526/10382. |
[27] |
C. Guan and Z. Yin, Global weak solutions for a two-component Camassa- Holm shallow water system, J. Funct. Anal., 260 (2011), 1132-1154.
doi: 10.1016/j.jfa.2010.11.015. |
[28] |
C. Guan and Z. Yin, Global weak solutions for a modified two-component Camassa-Holm equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 623-641.
doi: 10.1016/j.anihpc.2011.04.003. |
[29] |
C. Guan, H. He and Z. Yin, Well-posedness, blow-up phenomena and persistence properties for a two-component water wave system, Nonlinear Anal. Real World Appl., 25 (2015), 219-237.
doi: 10.1016/j.nonrwa.2015.04.001. |
[30] |
X. G. Geng and B. Xue, A three-component generalization of Camassa-Holm equation with N-peakon solutions, Adv. Math., 226 (2011), 827-839.
doi: 10.1016/j.aim.2010.07.009. |
[31] |
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. |
[32] |
D. Henry, Persistence properties for a family of nonlinear partial differential equations, Nonlinear Anal., 70 (2009), 1565-1573.
doi: 10.1016/j.na.2008.02.104. |
[33] |
D. D. Holm, L. Naraigh and C. Tronci, Singular solution of a modified twocomponent Camassa-Holm equation, Phys. Rev. E (3), 79 (2009), 016601, 13 pp.
doi: 10.1103/PhysRevE.79.016601. |
[34] |
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. |
[35] |
Y. Liu and Z. Yin, Global existence and blow-up phenomena for the Degasperis-Procesi equation, Comm. Math. Phys., 267 (2006), 801-820.
doi: 10.1007/s00220-006-0082-5. |
[36] |
W. Luo and Z. Yin, Global existence and local well-posedness for a three-component Camassa-Holm system with N-peakon solutions, J. Differential Equations, 259 (2015), 201-234.
doi: 10.1016/j.jde.2015.02.005. |
[37] |
G. Rodríguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46 (2001), 309-327.
doi: 10.1016/S0362-546X(01)00791-X. |
[38] |
W. Tan and Z. Yin, Global conservative solutions of a modified two-component Camassa-Holm shallow water system, J. Differential Equations, 251 (2011), 3558-3582.
doi: 10.1016/j.jde.2011.08.010. |
[39] |
W. Tan and Z. Yin, Global dissipative solutions of a modified two-component Camassa-Holm shallow water system, J. Math. Phys., 52 (2011), 033507, 24pp.
doi: 10.1063/1.3562928. |
[40] |
J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. |
[41] |
X. Wu and B. Guo, Persistence properties and infinite propagation for the modified 2-component Camassa-Holm equation, Discrete Contin. Dyn. Syst., 33 (2013), 3211-3223.
doi: 10.3934/dcds.2013.33.3211. |
[42] |
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. |
[43] |
K. Yan and Z. Yin, Well-posedness for a modified two-component Camassa-Holm system in critical spaces, Discrete Contin. Dyn. Syst., 33 (2013), 1699-1712.
doi: 10.3934/dcds.2013.33.1699. |
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