March  2015, 35(3): 871-889. doi: 10.3934/dcds.2015.35.871

On the Cauchy problem for a generalized Camassa-Holm equation

1. 

Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China

2. 

College of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan 453007, China

Received  March 2014 Revised  May 2014 Published  October 2014

In this paper, we consider the Cauchy problem for a generalized Camassa-Holm equation. We establish the local well-posedness in the Besov space $B_{2,1}^{3/2}$ and also prove that the local well-posedness fails in the Besov space $B_{2,\infty}^{3/2}$.
Citation: Defu Chen, Yongsheng Li, Wei Yan. On the Cauchy problem for a generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 871-889. doi: 10.3934/dcds.2015.35.871
References:
[1]

H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations,, Springer-Verlag, (2011).  doi: 10.1007/978-3-642-16830-7.  Google Scholar

[2]

G. R. Blanco, On the Cauchy problem for the Camassa-Holm equation,, Nonl. Anal. TMA., 46 (2001), 309.  doi: 10.1016/S0362-546X(01)00791-X.  Google Scholar

[3]

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

[4]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation,, Arch. Ration. Mech. Anal., 183 (2007), 215.  doi: 10.1007/s00205-006-0010-z.  Google Scholar

[5]

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

[6]

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

[7]

J. Y. Chemin, Perfect Incompressible Fluids,, Oxford University Press, (1998).   Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

A. Constantin and W. A. Strauss, Stability of peakons,, Comm. Pure Appl. Math., 53 (2000), 603.  doi: {10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L}.  Google Scholar

[19]

A. Constantin and W. A. Strauss, Stability of the Camassa-Holm solitons,, J. Nonl. Sci., 12 (2002), 415.  doi: 10.1007/s00332-002-0517-x.  Google Scholar

[20]

R. Danchin, A few remarks on the Camassa-Holm equation,, Diff. Integ. Eqns., 14 (2001), 953.   Google Scholar

[21]

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

[22]

R. Danchin and F. Fanelli, The well-posedness issue for the density-dependent Euler equations in endpoint Besov spaces,, J. Math. Pures Appl., 96 (2011), 253.  doi: 10.1016/j.matpur.2011.04.005.  Google Scholar

[23]

R. Danchin and P. B. Mucha, A critical functional framework for the inhomogeneous Navier-Stokes equations in the half-space,, J. Funct. Anal., 256 (2009), 881.  doi: 10.1016/j.jfa.2008.11.019.  Google Scholar

[24]

A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäklund transformations and hereditray symmetries,, Phys. D., 4 (1981), 47.  doi: 10.1016/0167-2789(81)90004-X.  Google Scholar

[25]

G. Gui and Y. Liu, On the Cauchy problem for the two-compenent Camassa-Holm system,, Math. Z., 268 (2011), 45.  doi: 10.1007/s00209-009-0660-2.  Google Scholar

[26]

A. A. Himonas and C. Holliman, The Cauchy problem for the Novikov equation,, Nonlinearity, 25 (2012), 449.  doi: 10.1088/0951-7715/25/2/449.  Google Scholar

[27]

A. A. Himonas and J. Holmes, Hölder continuity of the solution map for the Novikov equation,, J. Math. Phys., 54 (2013).  doi: 10.1063/1.4807729.  Google Scholar

[28]

A. A. Himonas and C. Holliman, The Cauchy problem for a Generalized Camassa-Holm equation,, Adv. Diff. Eqns., 19 (2014), 161.   Google Scholar

[29]

A. A. Himonas, G. Misiolek, G. Ponce and Y. Zhou, Persistence Properties and Unique Continuation of Solutions of the Camassa-Holm Equation,, Commun. Math. Phys., 271 (2007), 511.  doi: 10.1007/s00220-006-0172-4.  Google Scholar

[30]

H. Holden and X. Raynaud, A convergent numerical scheme for the Camassa-Holm equation based on multipeakons,, Discrete Continuous Dynam. Systems-A, 14 (2006), 505.   Google Scholar

[31]

H. Holden and X. Raynaud, Dissipative solutions for the Camassa-Holm equation,, Discrete Continuous Dynam. Systems-A, 24 (2009), 1047.  doi: 10.3934/dcds.2009.24.1047.  Google Scholar

[32]

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

[33]

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

[34]

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

[35]

L. Ni and Y. Zhou, Well-posedness and persistence properties for the Novikov equation,, J. Diff. Eqns., 250 (2011), 3002.  doi: 10.1016/j.jde.2011.01.030.  Google Scholar

[36]

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

[37]

J. F. Toland, Stokes waves,, Topol. Meth. Nonl., 7 (1996), 1.   Google Scholar

[38]

M. Vishik, Hydrodynamics in Besov spaces,, Arch. Rat. Mech. Anal., 145 (1998), 197.  doi: 10.1007/s002050050128.  Google Scholar

[39]

Z. Xin and P. Zhang, On the weak solutions to a shallow water equation,, Comm. Pure Appl. Math., 53 (2000), 1411.  doi: {10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5}.  Google Scholar

[40]

W. Yan, Y. Li and Y. Zhang, The Cauchy problem for the intergrable Novikov equation,, J. Diff. Eqns., 20 (2013), 1157.  doi: 10.1016/j.jde.2012.03.015.  Google Scholar

[41]

K. Yan and Z. Yin, On the Cauchy problem for a two-compnent Degasperis-Procesi system,, J. Diff. Eqns., 252 (2012), 2131.  doi: 10.1016/j.jde.2011.08.003.  Google Scholar

[42]

Y. Zhao, Y. Li and W. Yan, Local well-posedness and persistence property for the generalized Novikov equation,, Discrete Continuous Dynam. Systems-A, 34 (2014), 803.  doi: 10.3934/dcds.2014.34.803.  Google Scholar

show all references

References:
[1]

H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations,, Springer-Verlag, (2011).  doi: 10.1007/978-3-642-16830-7.  Google Scholar

[2]

G. R. Blanco, On the Cauchy problem for the Camassa-Holm equation,, Nonl. Anal. TMA., 46 (2001), 309.  doi: 10.1016/S0362-546X(01)00791-X.  Google Scholar

[3]

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

[4]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation,, Arch. Ration. Mech. Anal., 183 (2007), 215.  doi: 10.1007/s00205-006-0010-z.  Google Scholar

[5]

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

[6]

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

[7]

J. Y. Chemin, Perfect Incompressible Fluids,, Oxford University Press, (1998).   Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

A. Constantin and W. A. Strauss, Stability of peakons,, Comm. Pure Appl. Math., 53 (2000), 603.  doi: {10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L}.  Google Scholar

[19]

A. Constantin and W. A. Strauss, Stability of the Camassa-Holm solitons,, J. Nonl. Sci., 12 (2002), 415.  doi: 10.1007/s00332-002-0517-x.  Google Scholar

[20]

R. Danchin, A few remarks on the Camassa-Holm equation,, Diff. Integ. Eqns., 14 (2001), 953.   Google Scholar

[21]

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

[22]

R. Danchin and F. Fanelli, The well-posedness issue for the density-dependent Euler equations in endpoint Besov spaces,, J. Math. Pures Appl., 96 (2011), 253.  doi: 10.1016/j.matpur.2011.04.005.  Google Scholar

[23]

R. Danchin and P. B. Mucha, A critical functional framework for the inhomogeneous Navier-Stokes equations in the half-space,, J. Funct. Anal., 256 (2009), 881.  doi: 10.1016/j.jfa.2008.11.019.  Google Scholar

[24]

A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäklund transformations and hereditray symmetries,, Phys. D., 4 (1981), 47.  doi: 10.1016/0167-2789(81)90004-X.  Google Scholar

[25]

G. Gui and Y. Liu, On the Cauchy problem for the two-compenent Camassa-Holm system,, Math. Z., 268 (2011), 45.  doi: 10.1007/s00209-009-0660-2.  Google Scholar

[26]

A. A. Himonas and C. Holliman, The Cauchy problem for the Novikov equation,, Nonlinearity, 25 (2012), 449.  doi: 10.1088/0951-7715/25/2/449.  Google Scholar

[27]

A. A. Himonas and J. Holmes, Hölder continuity of the solution map for the Novikov equation,, J. Math. Phys., 54 (2013).  doi: 10.1063/1.4807729.  Google Scholar

[28]

A. A. Himonas and C. Holliman, The Cauchy problem for a Generalized Camassa-Holm equation,, Adv. Diff. Eqns., 19 (2014), 161.   Google Scholar

[29]

A. A. Himonas, G. Misiolek, G. Ponce and Y. Zhou, Persistence Properties and Unique Continuation of Solutions of the Camassa-Holm Equation,, Commun. Math. Phys., 271 (2007), 511.  doi: 10.1007/s00220-006-0172-4.  Google Scholar

[30]

H. Holden and X. Raynaud, A convergent numerical scheme for the Camassa-Holm equation based on multipeakons,, Discrete Continuous Dynam. Systems-A, 14 (2006), 505.   Google Scholar

[31]

H. Holden and X. Raynaud, Dissipative solutions for the Camassa-Holm equation,, Discrete Continuous Dynam. Systems-A, 24 (2009), 1047.  doi: 10.3934/dcds.2009.24.1047.  Google Scholar

[32]

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

[33]

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

[34]

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

[35]

L. Ni and Y. Zhou, Well-posedness and persistence properties for the Novikov equation,, J. Diff. Eqns., 250 (2011), 3002.  doi: 10.1016/j.jde.2011.01.030.  Google Scholar

[36]

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

[37]

J. F. Toland, Stokes waves,, Topol. Meth. Nonl., 7 (1996), 1.   Google Scholar

[38]

M. Vishik, Hydrodynamics in Besov spaces,, Arch. Rat. Mech. Anal., 145 (1998), 197.  doi: 10.1007/s002050050128.  Google Scholar

[39]

Z. Xin and P. Zhang, On the weak solutions to a shallow water equation,, Comm. Pure Appl. Math., 53 (2000), 1411.  doi: {10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5}.  Google Scholar

[40]

W. Yan, Y. Li and Y. Zhang, The Cauchy problem for the intergrable Novikov equation,, J. Diff. Eqns., 20 (2013), 1157.  doi: 10.1016/j.jde.2012.03.015.  Google Scholar

[41]

K. Yan and Z. Yin, On the Cauchy problem for a two-compnent Degasperis-Procesi system,, J. Diff. Eqns., 252 (2012), 2131.  doi: 10.1016/j.jde.2011.08.003.  Google Scholar

[42]

Y. Zhao, Y. Li and W. Yan, Local well-posedness and persistence property for the generalized Novikov equation,, Discrete Continuous Dynam. Systems-A, 34 (2014), 803.  doi: 10.3934/dcds.2014.34.803.  Google Scholar

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