-
Previous Article
Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system in $ \mathbb{R}^3 $ nonlinear KGS system
- CPAA Home
- This Issue
-
Next Article
Liouville type theorems for stable solutions of elliptic system involving the Grushin operator
Wave breaking phenomena and global existence for the weakly dissipative generalized Camassa-Holm equation
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 |
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.
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. |
[1] |
Xi Tu, Zhaoyang Yin. Local well-posedness and blow-up phenomena for a generalized Camassa-Holm equation with peakon solutions. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2781-2801. doi: 10.3934/dcds.2016.36.2781 |
[2] |
Zhaoyang Yin. Well-posedness and blow-up phenomena for the periodic generalized Camassa-Holm equation. Communications on Pure and Applied Analysis, 2004, 3 (3) : 501-508. doi: 10.3934/cpaa.2004.3.501 |
[3] |
Joachim Escher, Olaf Lechtenfeld, Zhaoyang Yin. Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2007, 19 (3) : 493-513. doi: 10.3934/dcds.2007.19.493 |
[4] |
Jinlu Li, Zhaoyang Yin. Well-posedness and blow-up phenomena for a generalized Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5493-5508. doi: 10.3934/dcds.2016042 |
[5] |
Ying Fu, Changzheng Qu, Yichen Ma. Well-posedness and blow-up phenomena for the interacting system of the Camassa-Holm and Degasperis-Procesi equations. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 1025-1035. doi: 10.3934/dcds.2010.27.1025 |
[6] |
Jae Min Lee, Stephen C. Preston. Local well-posedness of the Camassa-Holm equation on the real line. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3285-3299. doi: 10.3934/dcds.2017139 |
[7] |
Lei Zhang, Bin Liu. Well-posedness, blow-up criteria and gevrey regularity for a rotation-two-component camassa-holm system. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2655-2685. doi: 10.3934/dcds.2018112 |
[8] |
Min Zhu, Shuanghu Zhang. Blow-up of solutions to the periodic modified Camassa-Holm equation with varying linear dispersion. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7235-7256. doi: 10.3934/dcds.2016115 |
[9] |
Min Zhu, Ying Wang. Blow-up of solutions to the periodic generalized modified Camassa-Holm equation with varying linear dispersion. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 645-661. doi: 10.3934/dcds.2017027 |
[10] |
Wei Luo, Zhaoyang Yin. Local well-posedness in the critical Besov space and persistence properties for a three-component Camassa-Holm system with N-peakon solutions. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 5047-5066. doi: 10.3934/dcds.2016019 |
[11] |
Caixia Chen, Shu Wen. Wave breaking phenomena and global solutions for a generalized periodic two-component Camassa-Holm system. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3459-3484. doi: 10.3934/dcds.2012.32.3459 |
[12] |
Luigi Forcella, Kazumasa Fujiwara, Vladimir Georgiev, Tohru Ozawa. Local well-posedness and blow-up for the half Ginzburg-Landau-Kuramoto equation with rough coefficients and potential. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2661-2678. doi: 10.3934/dcds.2019111 |
[13] |
Wenxia Chen, Jingyi Liu, Danping Ding, Lixin Tian. Blow-up for two-component Camassa-Holm equation with generalized weak dissipation. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3769-3784. doi: 10.3934/cpaa.2020166 |
[14] |
Zhenhua Guo, Mina Jiang, Zhian Wang, Gao-Feng Zheng. Global weak solutions to the Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 883-906. doi: 10.3934/dcds.2008.21.883 |
[15] |
Min Zhu, Shuanghu Zhang. On the blow-up of solutions to the periodic modified integrable Camassa--Holm equation. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2347-2364. doi: 10.3934/dcds.2016.36.2347 |
[16] |
Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843 |
[17] |
Xinwei Yu, Zhichun Zhai. On the Lagrangian averaged Euler equations: local well-posedness and blow-up criterion. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1809-1823. doi: 10.3934/cpaa.2012.11.1809 |
[18] |
Wenjing Zhao. Local well-posedness and blow-up criteria of magneto-viscoelastic flows. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4637-4655. doi: 10.3934/dcds.2018203 |
[19] |
Tarek Saanouni. A note on global well-posedness and blow-up of some semilinear evolution equations. Evolution Equations and Control Theory, 2015, 4 (3) : 355-372. doi: 10.3934/eect.2015.4.355 |
[20] |
Shouming Zhou, Chunlai Mu. Global conservative and dissipative solutions of the generalized Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1713-1739. doi: 10.3934/dcds.2013.33.1713 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]