January  2017, 37(1): 645-661. doi: 10.3934/dcds.2017027

Blow-up of solutions to the periodic generalized modified Camassa-Holm equation with varying linear dispersion

1. 

Department of Mathematics, Nanjing Forestry University, Nanjing 210037, China

2. 

Department of Mathematics, University of Electronic Science and Technology of China, Chengdu 611731, China

Received  April 2016 Revised  May 2016 Published  November 2016

Considered herein is the blow-up mechanism to the periodic generalized modified Camassa-Holm equation with varying linear dispersion. The first one is designed for the case when linear dispersion is absent and derive a finite-time blow-up result. The key feature is the ratio between solution and its gradient. The second one handles the general situation when the weak linear dispersion is at present. Fortunately, there exist some conserved quantities that bound the $\|u_x\|_{L^4} $ for the periodic generalized modified Camassa-Holm equation, then the breakdown mechanisms are set up for the general case.

Citation: Min Zhu, Ying Wang. Blow-up of solutions to the periodic generalized modified Camassa-Holm equation with varying linear dispersion. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 645-661. doi: 10.3934/dcds.2017027
References:
[1]

L. Brandolese, Local-in-space criteria for blow-up in shallow water and dispersive rod equations, Comm. Math. Phys., 330 (2014), 401-414.  doi: 10.1007/s00220-014-1958-4.

[2]

L. Brandolese and M. F. Cortez, Blowup issues for a class of nonlinear dispersive wave equations, J. Differential Equations, 256 (2014), 3981-3998.  doi: 10.1016/j.jde.2014.03.008.

[3]

L. Brandolese and M. F. Cortez, On permanent and breading waves inn hyperelastic rods and rings, J. Funct. Anal., 266 (2014), 6954-6987.  doi: 10.1016/j.jfa.2014.02.039.

[4]

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.

[5]

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

[6]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.

[7]

M. ChenY. LiuC. Qu and S. Zhang, Oscillatio-induced blow-up to the modified Camassa-Holm equation with linear dispersion, Adv. Math., 272 (2015), 225-251.  doi: 10.1016/j.aim.2014.12.003.

[8]

K. S. Chou and C. Z. Qu, Integrable equations arising from motions of plane curves Ⅰ, Physica D, 162 (2002), 9-33.  doi: 10.1016/S0167-2789(01)00364-5.

[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, On the scattering problem for the Camassa-Holm equation, Proc. Roy. Soc. London A, 457 (2001), 953-970.  doi: 10.1098/rspa.2000.0701.

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

[13]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 26 (1998), 303-328. 

[14]

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.

[15]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math. (2), 173 (2011), 559-568.  doi: 10.4007/annals.2011.173.1.12.

[16]

A. Constantin and H. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804.  doi: 10.1007/s00014-003-0785-6.

[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 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.

[19]

A. Constantin and W. A. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610. 

[20]

A. Constantin and W. A. Strauss, Stability of a class of solitary waves in compressible elastic rods, Phys. Lett. A, 270 (2000), 140-148.  doi: 10.1016/S0375-9601(00)00255-3.

[21]

H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech., 127 (1998), 193-207.  doi: 10.1007/BF01170373.

[22]

A. Degasperis and M. Procesi, Asymptotic integrability, in Symmetry and perturbation theory (ed. A. Degasperis & G. Gaeta), pp 23-37, World Scientific, Singapore, 1999.

[23]

J. EscherY. Liu and Z. Yin, Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation, Indiana Univ. Math. J., 56 (2007), 87-117.  doi: 10.1512/iumj.2007.56.3040.

[24]

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.

[25]

B. Fuchssteiner and A. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D, 4 (1981/1982), 47-66.  doi: 10.1016/0167-2789(81)90004-X.

[26]

G. L. GuiY. LiuP. 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.

[27]

H. Holden and X. Raynaud, A convergent numerical scheme for the Camassa-Holm equation based on multipeakons, Discrete Contin. Dyn. Syst., 14 (2006), 505-523. 

[28]

S. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group, J. Math. Phys., 40 (1999), 857-868.  doi: 10.1063/1.532690.

[29]

J. Lenells, A Variational Approach to the Stability of Periodic Peakons, J. Nonlinear Math. Phys., 11 (2004), 151-163.  doi: 10.2991/jnmp.2004.11.2.2.

[30]

Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162 (2000), 27-63.  doi: 10.1006/jdeq.1999.3683.

[31]

Y. LiuP. OlverC. Qu and S. Zhang, On the blow-up of solutions to the integrable modified Camassa-Holm equation, Anal. Appl. (Singap.), 12 (2014), 355-368.  doi: 10.1142/S0219530514500274.

[32]

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.

[33]

G. Misołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208.  doi: 10.1016/S0393-0440(97)00010-7.

[34]

V. Novikov, Generalizations of the Camassa-Holm equation J. Phys. A 42 (2009), 342002, 14pp. doi: 10.1088/1751-8113/42/34/342002.

[35]

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

[36]

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

[37]

C. Z. QuX. C. Liu and Y. Liu, Stability of peakons for an integrable modified Camassa-Holm equation with cubic onlinearity, Comm. Math. Phys., 322 (2013), 967-997.  doi: 10.1007/s00220-013-1749-3.

[38]

C. Z. QuY. Fu and Y. Liu, Blow-up solutions and peakons to a generalized μ-Camassa-Holm integrable equation, Comm. Math. Phys., 311 (2014), 375-416.  doi: 10.1007/s00220-014-2007-z.

[39]

G. Rodriguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46 (2001), 309-327.  doi: 10.1016/S0362-546X(01)00791-X.

[40]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. 

[41]

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.

[42]

Z. Xin and P. Zhang, On the uniqueness and large time behavior of the weak solutions to a shallow water equation, Comm. Partial Differential Equations, 27 (2000), 1815-1844.  doi: 10.1081/PDE-120016129.

[43]

Z. Yin, On the blow-up of solutions of the periodic Camassa-Holm equation, Dyn. Cont. Discrete Impuls. Syst. Ser. A, Math. Anal., 12 (2005), 375-381. 

[44]

M. Zhu and S. Zhang, On the blow-up of solutions to the periodic modified integrable Camassa-Holm equation, Discrete Contin. Dyn. Syst., 36 (2016), 2347-2364.  doi: 10.3934/dcds.2016.36.2347.

show all references

References:
[1]

L. Brandolese, Local-in-space criteria for blow-up in shallow water and dispersive rod equations, Comm. Math. Phys., 330 (2014), 401-414.  doi: 10.1007/s00220-014-1958-4.

[2]

L. Brandolese and M. F. Cortez, Blowup issues for a class of nonlinear dispersive wave equations, J. Differential Equations, 256 (2014), 3981-3998.  doi: 10.1016/j.jde.2014.03.008.

[3]

L. Brandolese and M. F. Cortez, On permanent and breading waves inn hyperelastic rods and rings, J. Funct. Anal., 266 (2014), 6954-6987.  doi: 10.1016/j.jfa.2014.02.039.

[4]

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.

[5]

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

[6]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.

[7]

M. ChenY. LiuC. Qu and S. Zhang, Oscillatio-induced blow-up to the modified Camassa-Holm equation with linear dispersion, Adv. Math., 272 (2015), 225-251.  doi: 10.1016/j.aim.2014.12.003.

[8]

K. S. Chou and C. Z. Qu, Integrable equations arising from motions of plane curves Ⅰ, Physica D, 162 (2002), 9-33.  doi: 10.1016/S0167-2789(01)00364-5.

[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, On the scattering problem for the Camassa-Holm equation, Proc. Roy. Soc. London A, 457 (2001), 953-970.  doi: 10.1098/rspa.2000.0701.

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

[13]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 26 (1998), 303-328. 

[14]

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.

[15]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math. (2), 173 (2011), 559-568.  doi: 10.4007/annals.2011.173.1.12.

[16]

A. Constantin and H. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804.  doi: 10.1007/s00014-003-0785-6.

[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 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.

[19]

A. Constantin and W. A. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610. 

[20]

A. Constantin and W. A. Strauss, Stability of a class of solitary waves in compressible elastic rods, Phys. Lett. A, 270 (2000), 140-148.  doi: 10.1016/S0375-9601(00)00255-3.

[21]

H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech., 127 (1998), 193-207.  doi: 10.1007/BF01170373.

[22]

A. Degasperis and M. Procesi, Asymptotic integrability, in Symmetry and perturbation theory (ed. A. Degasperis & G. Gaeta), pp 23-37, World Scientific, Singapore, 1999.

[23]

J. EscherY. Liu and Z. Yin, Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation, Indiana Univ. Math. J., 56 (2007), 87-117.  doi: 10.1512/iumj.2007.56.3040.

[24]

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.

[25]

B. Fuchssteiner and A. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D, 4 (1981/1982), 47-66.  doi: 10.1016/0167-2789(81)90004-X.

[26]

G. L. GuiY. LiuP. 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.

[27]

H. Holden and X. Raynaud, A convergent numerical scheme for the Camassa-Holm equation based on multipeakons, Discrete Contin. Dyn. Syst., 14 (2006), 505-523. 

[28]

S. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group, J. Math. Phys., 40 (1999), 857-868.  doi: 10.1063/1.532690.

[29]

J. Lenells, A Variational Approach to the Stability of Periodic Peakons, J. Nonlinear Math. Phys., 11 (2004), 151-163.  doi: 10.2991/jnmp.2004.11.2.2.

[30]

Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162 (2000), 27-63.  doi: 10.1006/jdeq.1999.3683.

[31]

Y. LiuP. OlverC. Qu and S. Zhang, On the blow-up of solutions to the integrable modified Camassa-Holm equation, Anal. Appl. (Singap.), 12 (2014), 355-368.  doi: 10.1142/S0219530514500274.

[32]

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.

[33]

G. Misołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208.  doi: 10.1016/S0393-0440(97)00010-7.

[34]

V. Novikov, Generalizations of the Camassa-Holm equation J. Phys. A 42 (2009), 342002, 14pp. doi: 10.1088/1751-8113/42/34/342002.

[35]

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

[36]

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

[37]

C. Z. QuX. C. Liu and Y. Liu, Stability of peakons for an integrable modified Camassa-Holm equation with cubic onlinearity, Comm. Math. Phys., 322 (2013), 967-997.  doi: 10.1007/s00220-013-1749-3.

[38]

C. Z. QuY. Fu and Y. Liu, Blow-up solutions and peakons to a generalized μ-Camassa-Holm integrable equation, Comm. Math. Phys., 311 (2014), 375-416.  doi: 10.1007/s00220-014-2007-z.

[39]

G. Rodriguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46 (2001), 309-327.  doi: 10.1016/S0362-546X(01)00791-X.

[40]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. 

[41]

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.

[42]

Z. Xin and P. Zhang, On the uniqueness and large time behavior of the weak solutions to a shallow water equation, Comm. Partial Differential Equations, 27 (2000), 1815-1844.  doi: 10.1081/PDE-120016129.

[43]

Z. Yin, On the blow-up of solutions of the periodic Camassa-Holm equation, Dyn. Cont. Discrete Impuls. Syst. Ser. A, Math. Anal., 12 (2005), 375-381. 

[44]

M. Zhu and S. Zhang, On the blow-up of solutions to the periodic modified integrable Camassa-Holm equation, Discrete Contin. Dyn. Syst., 36 (2016), 2347-2364.  doi: 10.3934/dcds.2016.36.2347.

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