May  2015, 35(5): 2011-2039. doi: 10.3934/dcds.2015.35.2011

A note on the Cauchy problem of a modified Camassa-Holm equation with cubic nonlinearity

1. 

School of Mathematics, Northwest University, Shaanxi 710127, China

Received  June 2014 Revised  September 2014 Published  December 2014

Considered herein is the Cauchy problem for a modified Camassa-Holm equation with cubic nonlinearity. The local well-posedness in Besov space $B^s_{2,1}$ with the critical index $s=5/2$ is established. Then a lower bound for the maximal time of existence of its solutions is found. With analytic initial data, the solutions to this Cauchy problem are analytic in both variables, globally in space and locally in time, which extends the result of Himonas and Misiołek [A. Himonas, G. Misiołek, Analyticity of the Cauchy problem for an integrable evolution equation, Math. Ann. 327 (2003) 575---584] to more general $\mu$-version equations and systems.
Citation: Ying Fu. A note on the Cauchy problem of a modified Camassa-Holm equation with cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2011-2039. doi: 10.3934/dcds.2015.35.2011
References:
[1]

M. S. Baouendi and C. Goulaouic, Sharp estimates for analytic pseudodifferential operators and application to the Cauchy problems,, J. Differential Equations, 48 (1983), 241.  doi: 10.1016/0022-0396(83)90051-7.  Google Scholar

[2]

A. Boutet de Monvel, A. Kostenko, D. Shepelsky and G. Teschl, Long-time asymptotics for the Camassa-Holm equation,, SIAM J. Math. Anal., 41 (2009), 1559.  doi: 10.1137/090748500.  Google Scholar

[3]

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

[4]

J. Y. Chemin, Localization in Fourier space and Navier-Stokes system, Phase space analysis of partial differential equations,, Pubbl. Cent. Ric. Mat. Ennio Giorgi, I (2004), 53.   Google Scholar

[5]

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

[6]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach,, Ann. Inst. Fourier (Grenoble), 50 (2000), 321.  doi: 10.5802/aif.1757.  Google Scholar

[7]

A. Constantin, On the scattering problem for the Camassa-Holm equation,, Proc. Roy. Soc. Lond., 457 (2001), 953.  doi: 10.1098/rspa.2000.0701.  Google Scholar

[8]

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

[9]

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

[10]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation,, Ann. Scuola Norm. Sup. Pisa, 26 (1998), 303.   Google Scholar

[11]

A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation,, Math. Z., 233 (2000), 75.  doi: 10.1007/PL00004793.  Google Scholar

[12]

A. Constantin and J. Escher, Particle trajectories in solitary water waves,, Bull. Amer. Math. Soc., 44 (2007), 423.  doi: 10.1090/S0273-0979-07-01159-7.  Google Scholar

[13]

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

[14]

A. Constantin, V. S. Gerdjikov and R. I. Ivanov, Inverse scattering transform for the Camassa-Holm equation,, Inverse Problems, 22 (2006), 2197.  doi: 10.1088/0266-5611/22/6/017.  Google Scholar

[15]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Ration. Mech. Anal., 192 (2009), 165.  doi: 10.1007/s00205-008-0128-2.  Google Scholar

[16]

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

[17]

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

[18]

R. Danchin, A note on well-posedness for Camassa-Holm equation,, J. Differential Equations, 192 (2003), 429.  doi: 10.1016/S0022-0396(03)00096-2.  Google Scholar

[19]

A. Degasperis and M. Procesi, Asymptotic integrability,, Symmetry and perturbation theory (Rome 1998), (1999), 23.   Google Scholar

[20]

A. Degasperis, D. D. Holm and A. N. W. Hone, Integrable and non-integrable equations with peakons,, Nonlinear physics: Theory and experiment (Gallipoli 2002), II (2003), 37.  doi: 10.1142/9789812704467_0005.  Google Scholar

[21]

J. Escher, Y. Liu and Z. Yin, Global weak solutions and blow-up structure for the Degasperis-Procesi equation,, J. Funct. Anal., 241 (2006), 457.  doi: 10.1016/j.jfa.2006.03.022.  Google Scholar

[22]

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.  doi: 10.1512/iumj.2007.56.3040.  Google Scholar

[23]

Y. Fu, G. L. Gui, Y. Liu and C. Z. Qu, On the Cauchy problem for the integrable Camassa-Holm type equation with cubic nonlinearity,, J. Differential Equations, 255 (2013), 1905.  doi: 10.1016/j.jde.2013.05.024.  Google Scholar

[24]

Y. Fu, Y. Liu and C. Z. Qu, On the blow-up structure for the generalized periodic Camassa-Holm and Degasperis-Procesi equations,, J. Funct. Anal., 262 (2012), 3125.  doi: 10.1016/j.jfa.2012.01.009.  Google Scholar

[25]

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

[26]

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

[27]

A. A. Himonas and G. Misiołek, Analyticity of the Cauchy problem for an integrable evolution equation,, Math. Ann., 327 (2003), 575.  doi: 10.1007/s00208-003-0466-1.  Google Scholar

[28]

A. N. W. Hone and J. P. Wang, Integrable peakon equations with cubic nonlinearity,, J. Phys. A., 41 (2008).  doi: 10.1088/1751-8113/41/37/372002.  Google Scholar

[29]

A. N. W. Hone, H. Lundmark and J. Szmigielski, Explicit multipeakon solutions of Novikov's cubically nonlinearintegrable Camassa-Holm type equation,, Dyn. Partial Differ. Equ., 6 (2009), 253.  doi: 10.4310/DPDE.2009.v6.n3.a3.  Google Scholar

[30]

J. K. Hunter and R. Saxton, Dynamics of director fields,, SIAM J. Appl. Math., 51 (1991), 1498.  doi: 10.1137/0151075.  Google Scholar

[31]

R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid Mech., 455 (2002), 63.  doi: 10.1017/S0022112001007224.  Google Scholar

[32]

T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations,, in Spectral Theory and Differential Equations, 448 (1975), 25.   Google Scholar

[33]

B. Khesin, J. Lenells and G. Misiołek, Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms,, Math. Ann., 342 (2008), 617.  doi: 10.1007/s00208-008-0250-3.  Google Scholar

[34]

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

[35]

J. Lenells, G. Misiołek and F. Tiǧlay, Integrable evolution equations on spaces of tensor densities and their peakon solutions,, Comm. Math. Phys., 299 (2010), 129.  doi: 10.1007/s00220-010-1069-9.  Google Scholar

[36]

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.  doi: 10.1006/jdeq.1999.3683.  Google Scholar

[37]

Y. Liu and Z. Yin, Global existence and blow-up phenomena for the Degasperis-Procesi equation,, Comm. Math. Phys., 267 (2006), 801.  doi: 10.1007/s00220-006-0082-5.  Google Scholar

[38]

H. Lundmark, Formation and dynamics of shock waves in the Degasperis-Procesi equation,, J. Nonlinear Sci., 17 (2007), 169.  doi: 10.1007/s00332-006-0803-3.  Google Scholar

[39]

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

[40]

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

[41]

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

[42]

C. Z. Qu, Y. Fu and Y. Liu, Well-posedness, wave breaking and peakons for a modified $\mu$-Camassa-Holm equation,, J. Funct. Anal., 266 (2014), 433.  doi: 10.1016/j.jfa.2013.09.021.  Google Scholar

[43]

T. Schäfer and C. E. Wayne, Propagation of ultra-short optical pulses in cubic nonlinear media,, Physica D, 196 (2004), 90.  doi: 10.1016/j.physd.2004.04.007.  Google Scholar

[44]

J. Schiff, The Camassa-Holm equation: A loop group approach,, Physica D, 121 (1998), 24.  doi: 10.1016/S0167-2789(98)00099-2.  Google Scholar

[45]

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

[46]

F. Tiǧlay, The periodic Cauchy problem of the modified Hunter-Saxton equation,, J. Evol. Equ., 5 (2005), 509.  doi: 10.1007/s00028-005-0215-x.  Google Scholar

[47]

F. Tiǧlay, The periodic Cauchy problem for Novikov's equation,, Int. Math. Res. Not., 2011 (2011), 4633.  doi: 10.1093/imrn/rnq267.  Google Scholar

show all references

References:
[1]

M. S. Baouendi and C. Goulaouic, Sharp estimates for analytic pseudodifferential operators and application to the Cauchy problems,, J. Differential Equations, 48 (1983), 241.  doi: 10.1016/0022-0396(83)90051-7.  Google Scholar

[2]

A. Boutet de Monvel, A. Kostenko, D. Shepelsky and G. Teschl, Long-time asymptotics for the Camassa-Holm equation,, SIAM J. Math. Anal., 41 (2009), 1559.  doi: 10.1137/090748500.  Google Scholar

[3]

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

[4]

J. Y. Chemin, Localization in Fourier space and Navier-Stokes system, Phase space analysis of partial differential equations,, Pubbl. Cent. Ric. Mat. Ennio Giorgi, I (2004), 53.   Google Scholar

[5]

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

[6]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach,, Ann. Inst. Fourier (Grenoble), 50 (2000), 321.  doi: 10.5802/aif.1757.  Google Scholar

[7]

A. Constantin, On the scattering problem for the Camassa-Holm equation,, Proc. Roy. Soc. Lond., 457 (2001), 953.  doi: 10.1098/rspa.2000.0701.  Google Scholar

[8]

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

[9]

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

[10]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation,, Ann. Scuola Norm. Sup. Pisa, 26 (1998), 303.   Google Scholar

[11]

A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation,, Math. Z., 233 (2000), 75.  doi: 10.1007/PL00004793.  Google Scholar

[12]

A. Constantin and J. Escher, Particle trajectories in solitary water waves,, Bull. Amer. Math. Soc., 44 (2007), 423.  doi: 10.1090/S0273-0979-07-01159-7.  Google Scholar

[13]

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

[14]

A. Constantin, V. S. Gerdjikov and R. I. Ivanov, Inverse scattering transform for the Camassa-Holm equation,, Inverse Problems, 22 (2006), 2197.  doi: 10.1088/0266-5611/22/6/017.  Google Scholar

[15]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Ration. Mech. Anal., 192 (2009), 165.  doi: 10.1007/s00205-008-0128-2.  Google Scholar

[16]

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

[17]

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

[18]

R. Danchin, A note on well-posedness for Camassa-Holm equation,, J. Differential Equations, 192 (2003), 429.  doi: 10.1016/S0022-0396(03)00096-2.  Google Scholar

[19]

A. Degasperis and M. Procesi, Asymptotic integrability,, Symmetry and perturbation theory (Rome 1998), (1999), 23.   Google Scholar

[20]

A. Degasperis, D. D. Holm and A. N. W. Hone, Integrable and non-integrable equations with peakons,, Nonlinear physics: Theory and experiment (Gallipoli 2002), II (2003), 37.  doi: 10.1142/9789812704467_0005.  Google Scholar

[21]

J. Escher, Y. Liu and Z. Yin, Global weak solutions and blow-up structure for the Degasperis-Procesi equation,, J. Funct. Anal., 241 (2006), 457.  doi: 10.1016/j.jfa.2006.03.022.  Google Scholar

[22]

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.  doi: 10.1512/iumj.2007.56.3040.  Google Scholar

[23]

Y. Fu, G. L. Gui, Y. Liu and C. Z. Qu, On the Cauchy problem for the integrable Camassa-Holm type equation with cubic nonlinearity,, J. Differential Equations, 255 (2013), 1905.  doi: 10.1016/j.jde.2013.05.024.  Google Scholar

[24]

Y. Fu, Y. Liu and C. Z. Qu, On the blow-up structure for the generalized periodic Camassa-Holm and Degasperis-Procesi equations,, J. Funct. Anal., 262 (2012), 3125.  doi: 10.1016/j.jfa.2012.01.009.  Google Scholar

[25]

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

[26]

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

[27]

A. A. Himonas and G. Misiołek, Analyticity of the Cauchy problem for an integrable evolution equation,, Math. Ann., 327 (2003), 575.  doi: 10.1007/s00208-003-0466-1.  Google Scholar

[28]

A. N. W. Hone and J. P. Wang, Integrable peakon equations with cubic nonlinearity,, J. Phys. A., 41 (2008).  doi: 10.1088/1751-8113/41/37/372002.  Google Scholar

[29]

A. N. W. Hone, H. Lundmark and J. Szmigielski, Explicit multipeakon solutions of Novikov's cubically nonlinearintegrable Camassa-Holm type equation,, Dyn. Partial Differ. Equ., 6 (2009), 253.  doi: 10.4310/DPDE.2009.v6.n3.a3.  Google Scholar

[30]

J. K. Hunter and R. Saxton, Dynamics of director fields,, SIAM J. Appl. Math., 51 (1991), 1498.  doi: 10.1137/0151075.  Google Scholar

[31]

R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid Mech., 455 (2002), 63.  doi: 10.1017/S0022112001007224.  Google Scholar

[32]

T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations,, in Spectral Theory and Differential Equations, 448 (1975), 25.   Google Scholar

[33]

B. Khesin, J. Lenells and G. Misiołek, Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms,, Math. Ann., 342 (2008), 617.  doi: 10.1007/s00208-008-0250-3.  Google Scholar

[34]

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

[35]

J. Lenells, G. Misiołek and F. Tiǧlay, Integrable evolution equations on spaces of tensor densities and their peakon solutions,, Comm. Math. Phys., 299 (2010), 129.  doi: 10.1007/s00220-010-1069-9.  Google Scholar

[36]

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.  doi: 10.1006/jdeq.1999.3683.  Google Scholar

[37]

Y. Liu and Z. Yin, Global existence and blow-up phenomena for the Degasperis-Procesi equation,, Comm. Math. Phys., 267 (2006), 801.  doi: 10.1007/s00220-006-0082-5.  Google Scholar

[38]

H. Lundmark, Formation and dynamics of shock waves in the Degasperis-Procesi equation,, J. Nonlinear Sci., 17 (2007), 169.  doi: 10.1007/s00332-006-0803-3.  Google Scholar

[39]

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

[40]

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

[41]

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

[42]

C. Z. Qu, Y. Fu and Y. Liu, Well-posedness, wave breaking and peakons for a modified $\mu$-Camassa-Holm equation,, J. Funct. Anal., 266 (2014), 433.  doi: 10.1016/j.jfa.2013.09.021.  Google Scholar

[43]

T. Schäfer and C. E. Wayne, Propagation of ultra-short optical pulses in cubic nonlinear media,, Physica D, 196 (2004), 90.  doi: 10.1016/j.physd.2004.04.007.  Google Scholar

[44]

J. Schiff, The Camassa-Holm equation: A loop group approach,, Physica D, 121 (1998), 24.  doi: 10.1016/S0167-2789(98)00099-2.  Google Scholar

[45]

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

[46]

F. Tiǧlay, The periodic Cauchy problem of the modified Hunter-Saxton equation,, J. Evol. Equ., 5 (2005), 509.  doi: 10.1007/s00028-005-0215-x.  Google Scholar

[47]

F. Tiǧlay, The periodic Cauchy problem for Novikov's equation,, Int. Math. Res. Not., 2011 (2011), 4633.  doi: 10.1093/imrn/rnq267.  Google Scholar

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