November 2018, 38(11): 5523-5536. doi: 10.3934/dcds.2018243

Global existence for a two-component Camassa-Holm system with an arbitrary smooth function

1. 

School of Science, Wuhan University of Technology, Wuhan 430070, China

2. 

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

3. 

Faculty of Information Technology, Macau University of Science and Technology, Macau, China

* Corresponding author: Zhaoyang Yin

Received  October 2017 Revised  November 2017 Published  August 2018

Fund Project: Zhang is supported by NSFC (Grant No.: 11626177) and by the Fundamental Research Funds for the Central Universities (WUT: 2016IVA080). Yin was partially supported by NNSFC (No.11671407), FDCT (No. 098/2013/A3), Guangdong Special Support Program (No. 8-2015), and the key project of NSF of Guangdong province (No. 2016A030311004)

This paper is concerned with a two-component integrable Camassa-Holm type system with arbitrary smooth function $ H$. If the function $H$ belongs to a set $ \mathcal{H}$ (defined in Section 4), then we obtain the existence and uniqueness of global strong solutions and global weak solutions to the system. Our obtained results generalize and improve considerably recent results in [38,39].

Citation: Zeng Zhang, Zhaoyang Yin. Global existence for a two-component Camassa-Holm system with an arbitrary smooth function. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5523-5536. doi: 10.3934/dcds.2018243
References:
[1]

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.

[2]

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

[3]

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.

[4]

A. Constantin, Global existence of solutions and breaking waves for a shallow water equation: A geometric approach, Annales de l'Institut Fourier, 50 (2000), 321-362. doi: 10.5802/aif.1757.

[5]

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.

[6]

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

[7]

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.

[8]

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.

[9]

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.

[10]

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

[11]

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.

[12]

A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D.

[13]

A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61. doi: 10.1007/s002200050801.

[14]

A. Constantin and L. Molinet, Orbital stability of solitary waves for a shallow water equation, Phys. D, 157 (2001), 75-89. doi: 10.1016/S0167-2789(01)00298-6.

[15]

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.

[16]

R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988.

[17]

J. Escher and Z. Yin, Initial boundary value problems for nonlinear dispersive wave equations, J. Funct. Anal., 256 (2009), 479-508. doi: 10.1016/j.jfa.2008.07.010.

[18]

J. Escher and Z. Yin, Initial boundary value problems of the Camassa-Holm equation, Comm. Partial Differential Equations, 33 (2008), 377-395. doi: 10.1080/03605300701318872.

[19]

A. S. Fokas, On a class of physically important integrable equations, Phys. D, 87 (1995), 145-150. doi: 10.1016/0167-2789(95)00133-O.

[20]

Y. FuG. L. GuiY. Liu and C. Z. Qu, On the Cauchy problem for the integrable modified Camassa-Holm equation with cubic nonlinearity, J. Differential Equations, 255 (2013), 1905-1938. doi: 10.1016/j.jde.2013.05.024.

[21]

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

[22]

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

[23]

G. L. GuiY. LiuP. J. Olver and C. Z. Qu, Wave-breaking and peakons for a modified Camassa-Holm equation, Comm. Math. Phys., 319 (2013), 731-759. doi: 10.1007/s00220-012-1566-0.

[24]

H. Holden and X. Raynaud, Dissipative solutions for the Camassa-Holm equation, Discrete Contin. Dyn. Syst., 24 (2009), 1047-1112. doi: 10.3934/dcds.2009.24.1047.

[25]

Q. Hu and Z. Qiao, Analyticity, Gevrey regularity and unique continuation for an integrable multi-component peakon system with an arbitrary polynomial function, Discrete Contin. Dyn. Syst., 36 (2016), 6975-7000. doi: 10.3934/dcds.2016103.

[26]

J. Lenells, Conservation laws of the Camassa Holm equation, J. Phys. A, 38 (2005), 869-880. doi: 10.1088/0305-4470/38/4/007.

[27]

Y. Liu, Global existence and blow-up solutions for a nonlinear shallow water equation, Math. Ann., 335 (2006), 717-735. doi: 10.1007/s00208-006-0768-1.

[28]

J. Malek, J. Necas, M. Rokyta and M. Ruzicki, Weak and Measure-valued Solutions to Evolutionary PDEs, Chapman & Hall, 1996. doi: 10.1007/978-1-4899-6824-1.

[29]

P. J. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E (3), 53 (1996), 1900-1906. doi: 10.1103/PhysRevE.53.1900.

[30]

Z. Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons, J. Math. Phys., 47 (2006), 112701, 9 pp. doi: 10.1063/1.2365758.

[31]

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.

[32]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. doi: 10.12775/TMNA.1996.001.

[33]

B. XiaZ. Qiao and R. Zhou, A synthetical two-component model with peakon solutions, Stud. Appl. Math., 135 (2015), 248-276. doi: 10.1111/sapm.12085.

[34]

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.

[35]

K. YanZ. Qiao and Z. Yin, Qualitative analysis for a new integrable two-component Camassa-Holm system with peakon and weak kink solutions, Comm. Math. Phys, 336 (2015), 581-617. doi: 10.1007/s00220-014-2236-1.

[36]

Z. Zhang and Z. Yin, On the Cauchy problem for a four-component Camassa-Holm type system, Discrete Contin. Dyn. Syst., 35 (2015), 5153-5169. doi: 10.3934/dcds.2015.35.5153.

[37]

Z. Zhang and Z. Yin, Well-posedness, global existence and blow-up phenomena for an integrable multi-component Camassa- Holm system, Nonlinear Anal., 142 (2016), 112-133. doi: 10.1016/j.na.2016.04.004.

[38]

R. Zheng and Z. Yin, Blow-up phenomena and global existence for a two-component Camassa-Holm system with an arbitrary smooth function, Nonlinear Anal., 155 (2017), 176-185. doi: 10.1016/j.na.2017.02.004.

[39]

R. Zheng and Z. Yin, Global weak solutions for a two-component Camassa-Holm system with an arbitrary smooth function, Appl. Anal., (2017), 1-12. doi: 10.1080/00036811.2017.1350852.

show all references

References:
[1]

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.

[2]

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

[3]

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.

[4]

A. Constantin, Global existence of solutions and breaking waves for a shallow water equation: A geometric approach, Annales de l'Institut Fourier, 50 (2000), 321-362. doi: 10.5802/aif.1757.

[5]

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.

[6]

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

[7]

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.

[8]

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.

[9]

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.

[10]

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

[11]

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.

[12]

A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D.

[13]

A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61. doi: 10.1007/s002200050801.

[14]

A. Constantin and L. Molinet, Orbital stability of solitary waves for a shallow water equation, Phys. D, 157 (2001), 75-89. doi: 10.1016/S0167-2789(01)00298-6.

[15]

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.

[16]

R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988.

[17]

J. Escher and Z. Yin, Initial boundary value problems for nonlinear dispersive wave equations, J. Funct. Anal., 256 (2009), 479-508. doi: 10.1016/j.jfa.2008.07.010.

[18]

J. Escher and Z. Yin, Initial boundary value problems of the Camassa-Holm equation, Comm. Partial Differential Equations, 33 (2008), 377-395. doi: 10.1080/03605300701318872.

[19]

A. S. Fokas, On a class of physically important integrable equations, Phys. D, 87 (1995), 145-150. doi: 10.1016/0167-2789(95)00133-O.

[20]

Y. FuG. L. GuiY. Liu and C. Z. Qu, On the Cauchy problem for the integrable modified Camassa-Holm equation with cubic nonlinearity, J. Differential Equations, 255 (2013), 1905-1938. doi: 10.1016/j.jde.2013.05.024.

[21]

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

[22]

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

[23]

G. L. GuiY. LiuP. J. Olver and C. Z. Qu, Wave-breaking and peakons for a modified Camassa-Holm equation, Comm. Math. Phys., 319 (2013), 731-759. doi: 10.1007/s00220-012-1566-0.

[24]

H. Holden and X. Raynaud, Dissipative solutions for the Camassa-Holm equation, Discrete Contin. Dyn. Syst., 24 (2009), 1047-1112. doi: 10.3934/dcds.2009.24.1047.

[25]

Q. Hu and Z. Qiao, Analyticity, Gevrey regularity and unique continuation for an integrable multi-component peakon system with an arbitrary polynomial function, Discrete Contin. Dyn. Syst., 36 (2016), 6975-7000. doi: 10.3934/dcds.2016103.

[26]

J. Lenells, Conservation laws of the Camassa Holm equation, J. Phys. A, 38 (2005), 869-880. doi: 10.1088/0305-4470/38/4/007.

[27]

Y. Liu, Global existence and blow-up solutions for a nonlinear shallow water equation, Math. Ann., 335 (2006), 717-735. doi: 10.1007/s00208-006-0768-1.

[28]

J. Malek, J. Necas, M. Rokyta and M. Ruzicki, Weak and Measure-valued Solutions to Evolutionary PDEs, Chapman & Hall, 1996. doi: 10.1007/978-1-4899-6824-1.

[29]

P. J. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E (3), 53 (1996), 1900-1906. doi: 10.1103/PhysRevE.53.1900.

[30]

Z. Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons, J. Math. Phys., 47 (2006), 112701, 9 pp. doi: 10.1063/1.2365758.

[31]

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.

[32]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. doi: 10.12775/TMNA.1996.001.

[33]

B. XiaZ. Qiao and R. Zhou, A synthetical two-component model with peakon solutions, Stud. Appl. Math., 135 (2015), 248-276. doi: 10.1111/sapm.12085.

[34]

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.

[35]

K. YanZ. Qiao and Z. Yin, Qualitative analysis for a new integrable two-component Camassa-Holm system with peakon and weak kink solutions, Comm. Math. Phys, 336 (2015), 581-617. doi: 10.1007/s00220-014-2236-1.

[36]

Z. Zhang and Z. Yin, On the Cauchy problem for a four-component Camassa-Holm type system, Discrete Contin. Dyn. Syst., 35 (2015), 5153-5169. doi: 10.3934/dcds.2015.35.5153.

[37]

Z. Zhang and Z. Yin, Well-posedness, global existence and blow-up phenomena for an integrable multi-component Camassa- Holm system, Nonlinear Anal., 142 (2016), 112-133. doi: 10.1016/j.na.2016.04.004.

[38]

R. Zheng and Z. Yin, Blow-up phenomena and global existence for a two-component Camassa-Holm system with an arbitrary smooth function, Nonlinear Anal., 155 (2017), 176-185. doi: 10.1016/j.na.2017.02.004.

[39]

R. Zheng and Z. Yin, Global weak solutions for a two-component Camassa-Holm system with an arbitrary smooth function, Appl. Anal., (2017), 1-12. doi: 10.1080/00036811.2017.1350852.

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