December  2016, 36(12): 7235-7256. doi: 10.3934/dcds.2016115

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

1. 

Department of Mathematics, Nanjing Forestry University, Nanjing 210036

2. 

Department of Mathematics, Southwest University, Chongqing 400715

Received  December 2015 Revised  January 2016 Published  October 2016

Considered herein is the blow-up mechanism to the periodic modified Camassa-Holm equation with varying linear dispersion. We first consider 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. Using the continuity of the solutions and the right transformation, we then obtain this blow-up criterion to the case with negative linear dispersion and determine that the finite time blow-up can still occur if the initial momentum density is bounded below by the magnitude of the linear dispersion and the initial datum has a local mild-oscillation region. Finally, we demonstrate that when the linear dispersion is non-negative, formation of singularity can be induced by an initial datum with a sufficiently steep profile.
Citation: Min Zhu, Shuanghu Zhang. Blow-up of solutions to the periodic modified Camassa-Holm equation with varying linear dispersion. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7235-7256. doi: 10.3934/dcds.2016115
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 in 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. Chen, Y. Liu, C. 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., 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. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L.

[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), World Scientific, Singapore, (1999), 23-37.

[23]

J. Escher, Y. 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]

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

[25]

Y. Fu, G. L. Gui, Y. 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.

[26]

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

[27]

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

[28]

G. L. Gui, Y. Liu , P. 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.

[29]

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. doi: 10.3934/dcds.2006.14.505.

[30]

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.

[31]

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.

[32]

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.

[33]

Y. Liu, P. Olver, C. 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.

[34]

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.

[35]

Y.Matsuno, Smooth and singular multisolution solutions of a modified Camassa-Holm equation with cubic nonlinearity and linear dispersion, J. Phys. A., 47 (2014), 125203, 25pp. doi: 10.1088/1751-8113/47/12/125203.

[36]

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.

[37]

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

[38]

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.

[39]

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.

[40]

Z. Qiao and X. Q. Li, An integrable equation with nonsmooth solitons, Theor. Math. Phys., 167 (2011), 584-589. doi: 10.1007/s11232-011-0044-8.

[41]

C. Z. Qu, X. 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.

[42]

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.

[43]

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

[44]

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.

[45]

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 (2002), 1815-1844. doi: 10.1081/PDE-120016129.

[46]

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.

[47]

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 in 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. Chen, Y. Liu, C. 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., 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. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L.

[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), World Scientific, Singapore, (1999), 23-37.

[23]

J. Escher, Y. 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]

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

[25]

Y. Fu, G. L. Gui, Y. 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.

[26]

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

[27]

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

[28]

G. L. Gui, Y. Liu , P. 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.

[29]

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. doi: 10.3934/dcds.2006.14.505.

[30]

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.

[31]

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.

[32]

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.

[33]

Y. Liu, P. Olver, C. 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.

[34]

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.

[35]

Y.Matsuno, Smooth and singular multisolution solutions of a modified Camassa-Holm equation with cubic nonlinearity and linear dispersion, J. Phys. A., 47 (2014), 125203, 25pp. doi: 10.1088/1751-8113/47/12/125203.

[36]

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.

[37]

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

[38]

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.

[39]

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.

[40]

Z. Qiao and X. Q. Li, An integrable equation with nonsmooth solitons, Theor. Math. Phys., 167 (2011), 584-589. doi: 10.1007/s11232-011-0044-8.

[41]

C. Z. Qu, X. 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.

[42]

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.

[43]

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

[44]

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.

[45]

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 (2002), 1815-1844. doi: 10.1081/PDE-120016129.

[46]

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.

[47]

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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