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.

[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.

[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.

[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.

[5]

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

[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.

[7]

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

[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.

[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.

[10]

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

[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.

[12]

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

[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.

[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.

[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.

[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.

[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}.

[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}.

[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.

[20]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[28]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[37]

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

[38]

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

[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}.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[5]

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

[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.

[7]

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

[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.

[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.

[10]

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

[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.

[12]

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

[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.

[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.

[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.

[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.

[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}.

[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}.

[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.

[20]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[28]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[37]

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

[38]

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

[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}.

[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.

[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.

[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.

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