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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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