American Institute of Mathematical Sciences

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September  2016, 36(9): 5047-5066. doi: 10.3934/dcds.2016019

Local well-posedness in the critical Besov space and persistence properties for a three-component Camassa-Holm system with N-peakon solutions

Wei Luo 1, and  Zhaoyang Yin 2,

1. 

Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China
2. 

Department of Mathematics, Zhongshan University, Guangzhou, 510275

Received  July 2015 Revised  November 2015 Published  May 2016

In this paper we mainly investigate the Cauchy problem of a three-component Camassa-Holm system. By using Littlewood-Paley theory and transport equations theory, we establish the local well-posedness of the system in the critical Besov space. Moreover, we obtain some weighted $L^p$ estimates of strong solutions to the system. By taking suitable weighted functions, we can get the persistence properties of strong solutions on exponential, algebraic and logarithmic decay rates, respectively.
Keywords: critical Besov space, local well-posedness, persistence properties., A three-component Camassa-Holm system.
Mathematics Subject Classification: Primary: 35Q53; Secondary: 35A01, 35B44, 35B6.
Citation: Wei Luo, Zhaoyang Yin. Local well-posedness in the critical Besov space and persistence properties for a three-component Camassa-Holm system with N-peakon solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5047-5066. doi: 10.3934/dcds.2016019
References:
[1]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations,, Grundlehren der Mathematischen Wissenschaften, (2011).  doi: 10.1007/978-3-642-16830-7.  Google Scholar

[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.   Google Scholar

[3]

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

[5]

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

[6]

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

[7]

A. Constantin, The Hamiltonian structure of the Camassa-Holm equation,, Exposition. Math., 15 (1997), 53.   Google Scholar

[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.  doi: 10.1098/rspa.2000.0701.  Google Scholar

[9]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach,, Ann. Inst. Fourier (Grenoble), 50 (2000), 321.  doi: 10.5802/aif.1757.  Google Scholar

[10]

A. Constantin, Finite propagation speed for the Camassa-Holm equation,, J. Math. Phys., 46 (2005).  doi: 10.1063/1.1845603.  Google Scholar

[11]

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

[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.   Google Scholar

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

[14]

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

[15]

A. Constantin and J. Escher, Particle trajectories in solitary water waves,, Bull. Amer. Math. Soc., 44 (2007), 423.  doi: 10.1090/S0273-0979-07-01159-7.  Google Scholar

[16]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. of Math., 173 (2011), 559.  doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[17]

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

[18]

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

[19]

A. Constantin and W. A. Strauss, Stability of peakons,, Comm. Pure Appl. Math., 53 (2000), 603.  doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L.  Google Scholar

[20]

A. Constantin and R. Ivanov, On an integrable two-component Camassa-Holm shallow water system,, Phys. Lett. A, 372 (2008), 7129.  doi: 10.1016/j.physleta.2008.10.050.  Google Scholar

[21]

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

[22]

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

[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.  doi: 10.3934/dcds.2007.19.493.  Google Scholar

[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.  Google Scholar

[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.  doi: 10.1016/j.jde.2009.08.002.  Google Scholar

[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.  doi: 10.1090/conm/526/10382.  Google Scholar

[27]

C. Guan and Z. Yin, Global weak solutions for a two-component Camassa- Holm shallow water system,, J. Funct. Anal., 260 (2011), 1132.  doi: 10.1016/j.jfa.2010.11.015.  Google Scholar

[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.  doi: 10.1016/j.anihpc.2011.04.003.  Google Scholar

[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.  doi: 10.1016/j.nonrwa.2015.04.001.  Google Scholar

[30]

X. G. Geng and B. Xue, A three-component generalization of Camassa-Holm equation with N-peakon solutions,, Adv. Math., 226 (2011), 827.  doi: 10.1016/j.aim.2010.07.009.  Google Scholar

[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.  doi: 10.1016/j.jfa.2010.02.008.  Google Scholar

[32]

D. Henry, Persistence properties for a family of nonlinear partial differential equations,, Nonlinear Anal., 70 (2009), 1565.  doi: 10.1016/j.na.2008.02.104.  Google Scholar

[33]

D. D. Holm, L. Naraigh and C. Tronci, Singular solution of a modified twocomponent Camassa-Holm equation,, Phys. Rev. E (3), 79 (2009).  doi: 10.1103/PhysRevE.79.016601.  Google Scholar

[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.  doi: 10.1007/s00220-006-0172-4.  Google Scholar

[35]

Y. Liu and Z. Yin, Global existence and blow-up phenomena for the Degasperis-Procesi equation,, Comm. Math. Phys., 267 (2006), 801.  doi: 10.1007/s00220-006-0082-5.  Google Scholar

[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.  doi: 10.1016/j.jde.2015.02.005.  Google Scholar

[37]

G. Rodríguez-Blanco, On the Cauchy problem for the Camassa-Holm equation,, Nonlinear Anal., 46 (2001), 309.  doi: 10.1016/S0362-546X(01)00791-X.  Google Scholar

[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.  doi: 10.1016/j.jde.2011.08.010.  Google Scholar

[39]

W. Tan and Z. Yin, Global dissipative solutions of a modified two-component Camassa-Holm shallow water system,, \emph{J. Math. Phys.}, 52 (2011).  doi: 10.1063/1.3562928.  Google Scholar

[40]

J. F. Toland, Stokes waves,, Topol. Methods Nonlinear Anal., 7 (1996), 1.   Google Scholar

[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.  doi: 10.3934/dcds.2013.33.3211.  Google Scholar

[42]

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

[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.  doi: 10.3934/dcds.2013.33.1699.  Google Scholar

show all references

References:
[1]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations,, Grundlehren der Mathematischen Wissenschaften, (2011).  doi: 10.1007/978-3-642-16830-7.  Google Scholar

[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.   Google Scholar

[3]

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

[4]

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

[5]

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

[6]

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

[7]

A. Constantin, The Hamiltonian structure of the Camassa-Holm equation,, Exposition. Math., 15 (1997), 53.   Google Scholar

[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.  doi: 10.1098/rspa.2000.0701.  Google Scholar

[9]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach,, Ann. Inst. Fourier (Grenoble), 50 (2000), 321.  doi: 10.5802/aif.1757.  Google Scholar

[10]

A. Constantin, Finite propagation speed for the Camassa-Holm equation,, J. Math. Phys., 46 (2005).  doi: 10.1063/1.1845603.  Google Scholar

[11]

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

[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.   Google Scholar

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

[14]

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

[15]

A. Constantin and J. Escher, Particle trajectories in solitary water waves,, Bull. Amer. Math. Soc., 44 (2007), 423.  doi: 10.1090/S0273-0979-07-01159-7.  Google Scholar

[16]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. of Math., 173 (2011), 559.  doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[17]

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

[18]

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

[19]

A. Constantin and W. A. Strauss, Stability of peakons,, Comm. Pure Appl. Math., 53 (2000), 603.  doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L.  Google Scholar

[20]

A. Constantin and R. Ivanov, On an integrable two-component Camassa-Holm shallow water system,, Phys. Lett. A, 372 (2008), 7129.  doi: 10.1016/j.physleta.2008.10.050.  Google Scholar

[21]

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

[22]

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

[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.  doi: 10.3934/dcds.2007.19.493.  Google Scholar

[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.  Google Scholar

[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.  doi: 10.1016/j.jde.2009.08.002.  Google Scholar

[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.  doi: 10.1090/conm/526/10382.  Google Scholar

[27]

C. Guan and Z. Yin, Global weak solutions for a two-component Camassa- Holm shallow water system,, J. Funct. Anal., 260 (2011), 1132.  doi: 10.1016/j.jfa.2010.11.015.  Google Scholar

[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.  doi: 10.1016/j.anihpc.2011.04.003.  Google Scholar

[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.  doi: 10.1016/j.nonrwa.2015.04.001.  Google Scholar

[30]

X. G. Geng and B. Xue, A three-component generalization of Camassa-Holm equation with N-peakon solutions,, Adv. Math., 226 (2011), 827.  doi: 10.1016/j.aim.2010.07.009.  Google Scholar

[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.  doi: 10.1016/j.jfa.2010.02.008.  Google Scholar

[32]

D. Henry, Persistence properties for a family of nonlinear partial differential equations,, Nonlinear Anal., 70 (2009), 1565.  doi: 10.1016/j.na.2008.02.104.  Google Scholar

[33]

D. D. Holm, L. Naraigh and C. Tronci, Singular solution of a modified twocomponent Camassa-Holm equation,, Phys. Rev. E (3), 79 (2009).  doi: 10.1103/PhysRevE.79.016601.  Google Scholar

[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.  doi: 10.1007/s00220-006-0172-4.  Google Scholar

[35]

Y. Liu and Z. Yin, Global existence and blow-up phenomena for the Degasperis-Procesi equation,, Comm. Math. Phys., 267 (2006), 801.  doi: 10.1007/s00220-006-0082-5.  Google Scholar

[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.  doi: 10.1016/j.jde.2015.02.005.  Google Scholar

[37]

G. Rodríguez-Blanco, On the Cauchy problem for the Camassa-Holm equation,, Nonlinear Anal., 46 (2001), 309.  doi: 10.1016/S0362-546X(01)00791-X.  Google Scholar

[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.  doi: 10.1016/j.jde.2011.08.010.  Google Scholar

[39]

W. Tan and Z. Yin, Global dissipative solutions of a modified two-component Camassa-Holm shallow water system,, \emph{J. Math. Phys.}, 52 (2011).  doi: 10.1063/1.3562928.  Google Scholar

[40]

J. F. Toland, Stokes waves,, Topol. Methods Nonlinear Anal., 7 (1996), 1.   Google Scholar

[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.  doi: 10.3934/dcds.2013.33.3211.  Google Scholar

[42]

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

[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.  doi: 10.3934/dcds.2013.33.1699.  Google Scholar

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