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February  2022, 21(2): 555-566. doi: 10.3934/cpaa.2021188

Wave breaking phenomena and global existence for the weakly dissipative generalized Camassa-Holm equation

Yonghui Zhou 1,2, and  Shuguan Ji 1,,

1. 

School of Mathematics and Statistics, and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, 130024, China
2. 

School of Mathematics and Statistics, Hexi University, Zhangye 734000, China

* Corresponding author

Received  July 2021 Revised  September 2021 Published  February 2022 Early access  November 2021

Fund Project: This work is partially supported by NSFC Grants (nos. 12071065 and 11871140) and the National Key Research and Development Program of China (nos. 2020YFA0713602 and 2020YFC1808301)

In this paper, we mainly study several problems on the weakly dissipative generalized Camassa-Holm equation. We first establish the local well-posedness of solutions by Kato's semigroup theory. We then derive the necessary and sufficient condition of the blow-up of solutions and a criteria to guarantee occurrence of wave breaking. Moreover, when the solution blows up, we obtain the precise blow-up rate. We finally show that the equation has a unique global solution provided the momentum density associated with their initial datum satisfies appropriate sign conditions.

Keywords: Camassa-Holm equation, local well-posedness, wave breaking, blow-up rate, global solutions.
Mathematics Subject Classification: Primary: 35A01, 35B44, 35Q35.
Citation: Yonghui Zhou, Shuguan Ji. Wave breaking phenomena and global existence for the weakly dissipative generalized Camassa-Holm equation. Communications on Pure and Applied Analysis, 2022, 21 (2) : 555-566. doi: 10.3934/cpaa.2021188
References:
[1]

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.

[2]

R. Camassa and D. Holm, An integrable shallow water equation with peaked soliton, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.

[3]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 18 (1998), 229-243.  doi: 10.1007/BF02392586.

[4]

A. Constantin and J. Escher, Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Commun. Pure Appl. Math., 51 (1998), 475-504.  doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5.

[5]

A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Commun. Math. Phys., 211 (2000), 45-61.  doi: 10.1007/s002200050801.

[6]

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.

[7]

A. Constantin and W. A. Strauss, Stability of peakons, Commun. Pure Appl. Math., 53 (2000), 603-610.  doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L.

[8]

A. Constantin and L. Molinet, The initial value problem for a generalized Boussinesq equation, Differ. Integral Equ., 15 (2002), 1061-1072. 

[9]

A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries, Phys. D, 4 (1981), 47-66.  doi: 10.1016/0167-2789(81)90004-X.

[10]

I. L. Freire, N. S. Filho, L. C. Souza and C. E. Toffoli, Invariants and wave breaking analysis of a Camassa-Holm type equation with quadratic and cubic non-linearities, J. Differ. Equ., 269 (2020), 56-77.  doi: 10.1016/j.jde.2020.04.041.

[11]

J. M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time, J. Differ. Equ., 74 (1988), 369-390.  doi: 10.1016/0022-0396(88)90010-1.

[12]

Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differ. Equ., 162 (2000), 27-63.  doi: 10.1006/jdeq.1999.3683.

[13]

T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in Spectral Theory and Differential Equations, Lecture Notes in Math, Springer, Berlin, 1975. doi: 10.1007/BFB0067080.

[14]

T. Kato and G. Ponce, Commutator estimate and the Eular and Navier-Stokes equations, Commun. Pure. Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.

[15]

E. Ott and R. N. Sudan, Damping of solitary waves, Phys. Fluids, 13 (1970), 1432-1434.  doi: 10.1063/1.1693097.

[16]

E. Wahlén, Global existence of weak solutions to the Camassa-Holm equation, Int. Math. Res. Not., 2006 (2006), 1-12.  doi: 10.1155/IMRN/2006/28976.

[17]

S. Wu and Z. Yin, Global existence and blow up phenomena for the weakly disspative Camassa-Holm equation, J. Differ. Equ., 246 (2009), 4309-4321.  doi: 10.1016/j.jde.2008.12.008.

[18]

X. Wu and B. Guo, The Cauchy problem of the modified CH and DP equations, IMA J. Appl. Math., 80 (2015), 906-930.  doi: 10.1093/imamat/hxu032.

[19]

Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Commun. Pure Appl. Math., 53 (2000), 1411-1433.  doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5.

[20]

Z. Yin, On the Blow-up scenario for the generalized Camassa-Holm equation, Commun. Partial Differ. Equ., 29 (2004), 867-877.  doi: 10.1081/PDE-120037334.

[21]

Z. Yin, Well-posedness and blow-up phenomena for the periodic generalized Camassa-Holm equation, Commun. Pure Appl. Anal., 3 (2004), 501-508.  doi: 10.3934/cpaa.2004.3.501.

[22]

Z. Yin, On the Cauchy problem for the generalized Camassa-Holm equation, Nonlinear Anal., 66 (2007), 460-471.  doi: 10.1016/j.na.2005.11.040.

show all references

References:
[1]

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.

[2]

R. Camassa and D. Holm, An integrable shallow water equation with peaked soliton, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.

[3]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 18 (1998), 229-243.  doi: 10.1007/BF02392586.

[4]

A. Constantin and J. Escher, Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Commun. Pure Appl. Math., 51 (1998), 475-504.  doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5.

[5]

A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Commun. Math. Phys., 211 (2000), 45-61.  doi: 10.1007/s002200050801.

[6]

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.

[7]

A. Constantin and W. A. Strauss, Stability of peakons, Commun. Pure Appl. Math., 53 (2000), 603-610.  doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L.

[8]

A. Constantin and L. Molinet, The initial value problem for a generalized Boussinesq equation, Differ. Integral Equ., 15 (2002), 1061-1072. 

[9]

A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries, Phys. D, 4 (1981), 47-66.  doi: 10.1016/0167-2789(81)90004-X.

[10]

I. L. Freire, N. S. Filho, L. C. Souza and C. E. Toffoli, Invariants and wave breaking analysis of a Camassa-Holm type equation with quadratic and cubic non-linearities, J. Differ. Equ., 269 (2020), 56-77.  doi: 10.1016/j.jde.2020.04.041.

[11]

J. M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time, J. Differ. Equ., 74 (1988), 369-390.  doi: 10.1016/0022-0396(88)90010-1.

[12]

Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differ. Equ., 162 (2000), 27-63.  doi: 10.1006/jdeq.1999.3683.

[13]

T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in Spectral Theory and Differential Equations, Lecture Notes in Math, Springer, Berlin, 1975. doi: 10.1007/BFB0067080.

[14]

T. Kato and G. Ponce, Commutator estimate and the Eular and Navier-Stokes equations, Commun. Pure. Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.

[15]

E. Ott and R. N. Sudan, Damping of solitary waves, Phys. Fluids, 13 (1970), 1432-1434.  doi: 10.1063/1.1693097.

[16]

E. Wahlén, Global existence of weak solutions to the Camassa-Holm equation, Int. Math. Res. Not., 2006 (2006), 1-12.  doi: 10.1155/IMRN/2006/28976.

[17]

S. Wu and Z. Yin, Global existence and blow up phenomena for the weakly disspative Camassa-Holm equation, J. Differ. Equ., 246 (2009), 4309-4321.  doi: 10.1016/j.jde.2008.12.008.

[18]

X. Wu and B. Guo, The Cauchy problem of the modified CH and DP equations, IMA J. Appl. Math., 80 (2015), 906-930.  doi: 10.1093/imamat/hxu032.

[19]

Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Commun. Pure Appl. Math., 53 (2000), 1411-1433.  doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5.

[20]

Z. Yin, On the Blow-up scenario for the generalized Camassa-Holm equation, Commun. Partial Differ. Equ., 29 (2004), 867-877.  doi: 10.1081/PDE-120037334.

[21]

Z. Yin, Well-posedness and blow-up phenomena for the periodic generalized Camassa-Holm equation, Commun. Pure Appl. Anal., 3 (2004), 501-508.  doi: 10.3934/cpaa.2004.3.501.

[22]

Z. Yin, On the Cauchy problem for the generalized Camassa-Holm equation, Nonlinear Anal., 66 (2007), 460-471.  doi: 10.1016/j.na.2005.11.040.

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