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doi: 10.3934/cpaa.2021188
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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 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 & Applied Analysis, 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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