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

Min  Zhu; Shuanghu  Zhang

doi:10.3934/dcds.2016115

Article Contents

2016, Volume 36, Issue 12: 7235-7256. Doi: 10.3934/dcds.2016115

This issue Previous Article A powered Gronwall-type inequality and applications to stochastic differential equations Next Article

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

Min Zhu^1, and
Shuanghu Zhang^2,

1.
Department of Mathematics, Nanjing Forestry University, Nanjing 210036
2.
Department of Mathematics, Southwest University, Chongqing 400715

Received: November 30, 2015

Revised: December 31, 2015

Published: May 2017

Abstract Related Papers Cited by

Abstract

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.

Keywords:

Periodic modified Camassa-Holm equation,
blow up,
integrable equation,
sign-changing momentum,
varying linear dispersion.

Mathematics Subject Classification: 35B44, 35G25.

Citation:

Related Papers

Cited by

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 (1981/1982), 47-66. 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.