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

Ying Fu

doi:10.3934/dcds.2015.35.2011

Article Contents

2015, Volume 35, Issue 5: 2011-2039. Doi: 10.3934/dcds.2015.35.2011

This issue Previous Article Integrability of potentials of degree $k \neq \pm 2$. Second order variational equations between Kolchin solvability and Abelianity Next Article Blow-up for the two-component Camassa--Holm system

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

Ying Fu^1,

1.
School of Mathematics, Northwest University, Shaanxi 710127

Received: May 31, 2014

Revised: August 31, 2014

Published: May 2017

Abstract Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 35B30, 35G25; Secondary: 35L05.

Citation:

Related Papers

Cited by

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), 241-268.doi: 10.1016/0022-0396(83)90051-7.
[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), 1559-1588.doi: 10.1137/090748500.
[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.
[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, CRM series, Pisa, I (2004), 53-135.
[5]	K. S. Chou and C. Z. Qu, Integrable equations arising from motions of plane curves, Physica D, 162 (2002), 9-33.doi: 10.1016/S0167-2789(01)00364-5.
[6]	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.
[7]	A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. Roy. Soc. Lond., 457 (2001), 953-970.doi: 10.1098/rspa.2000.0701.
[8]	A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.doi: 10.1007/s00222-006-0002-5.
[9]	A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.doi: 10.1007/BF02392586.
[10]	A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa, 26 (1998), 303-328.
[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), 75-91.doi: 10.1007/PL00004793.
[12]	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.
[13]	A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. Math., 173 (2011), 559-568.doi: 10.4007/annals.2011.173.1.12.
[14]	A. Constantin, V. S. Gerdjikov and R. I. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207.doi: 10.1088/0266-5611/22/6/017.
[15]	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.
[16]	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.
[17]	R. Danchin, A few remarks on the Camassa-Holm equation, Differential and Integral Equations, 14 (2001), 953-988.
[18]	R. Danchin, A note on well-posedness for Camassa-Holm equation, J. Differential Equations, 192 (2003), 429-444.doi: 10.1016/S0022-0396(03)00096-2.
[19]	A. Degasperis and M. Procesi, Asymptotic integrability, Symmetry and perturbation theory (Rome 1998), World Sci. Publ., River Edge, NJ, (1999), 23-37.
[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), World Sci. Publ., River Edge, NJ, II (2003), 37-43.doi: 10.1142/9789812704467_0005.
[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), 457-485.doi: 10.1016/j.jfa.2006.03.022.
[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), 87-117.doi: 10.1512/iumj.2007.56.3040.
[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), 1905-1938.doi: 10.1016/j.jde.2013.05.024.
[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), 3125-3158.doi: 10.1016/j.jfa.2012.01.009.
[25]	B. Fuchssteiner and A. S. 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. 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), 731-759.doi: 10.1007/s00220-012-1566-0.
[27]	A. A. Himonas and G. Misiołek, Analyticity of the Cauchy problem for an integrable evolution equation, Math. Ann., 327 (2003), 575-584.doi: 10.1007/s00208-003-0466-1.
[28]	A. N. W. Hone and J. P. Wang, Integrable peakon equations with cubic nonlinearity, J. Phys. A., 41 (2008), 372002, 10 pp.doi: 10.1088/1751-8113/41/37/372002.
[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), 253-289.doi: 10.4310/DPDE.2009.v6.n3.a3.
[30]	J. K. Hunter and R. Saxton, Dynamics of director fields, SIAM J. Appl. Math., 51 (1991), 1498-1521.doi: 10.1137/0151075.
[31]	R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82.doi: 10.1017/S0022112001007224.
[32]	T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in Spectral Theory and Differential Equations, Lecture Notes in Math., 448, Springer Verlag, Berlin, (1975), 25-70.
[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), 617-656.doi: 10.1007/s00208-008-0250-3.
[34]	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.
[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), 129-161.doi: 10.1007/s00220-010-1069-9.
[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), 27-63.doi: 10.1006/jdeq.1999.3683.
[37]	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.
[38]	H. Lundmark, Formation and dynamics of shock waves in the Degasperis-Procesi equation, J. Nonlinear Sci., 17 (2007), 169-198.doi: 10.1007/s00332-006-0803-3.
[39]	G. Misiołek, 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.
[40]	V. Novikov, Generalizations of the Camassa-Holm equation, J. Phys. A., 42 (2009), 342002, 14 pp.doi: 10.1088/1751-8113/42/34/342002.
[41]	P. J. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53 (1996), 1900-1906.doi: 10.1103/PhysRevE.53.1900.
[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), 433-477.doi: 10.1016/j.jfa.2013.09.021.
[43]	T. Schäfer and C. E. Wayne, Propagation of ultra-short optical pulses in cubic nonlinear media, Physica D, 196 (2004), 90-105.doi: 10.1016/j.physd.2004.04.007.
[44]	J. Schiff, The Camassa-Holm equation: A loop group approach, Physica D, 121 (1998), 24-43.doi: 10.1016/S0167-2789(98)00099-2.
[45]	J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.
[46]	F. Tiǧlay, The periodic Cauchy problem of the modified Hunter-Saxton equation, J. Evol. Equ., 5 (2005), 509-527.doi: 10.1007/s00028-005-0215-x.
[47]	F. Tiǧlay, The periodic Cauchy problem for Novikov's equation, Int. Math. Res. Not., 2011 (2011), 4633-4648.doi: 10.1093/imrn/rnq267.