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Large spectral gap induced by small delay and its application to reduction
On the blow up solutions to a two-component cubic Camassa-Holm system with peakons
| 1. | School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China |
| 2. | Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, China |
This paper is concerned with the Cauchy problem for a two- component cubic Camassa-Holm system with peakons, which is a natural multi-component extension of the single Fokas-Olver-Rosenau-Qiao equation. By sufficiently exploiting the fine structure of the system, we derive two useful conservation laws which turns out an exponential increase estimate for the $ L^\infty $-norm of the strong solution within its lifespan. As a result, two new blow-up solutions with certain initial profiles are established.
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. D. Holm,
An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
| [3] |
R. M. Chen, F. Gao, Y. Liu and C. Qu,
Analysis on the blow-up of solutions to a class of integrable peakon equations, J. Funct. Anal., 270 (2016), 2343-2374.
doi: 10.1016/j.jfa.2016.01.017. |
| [4] |
A. Constantin,
Global existence of solutions and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.
doi: 10.5802/aif.1757. |
| [5] |
A. Constantin,
The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.
doi: 10.1007/s00222-006-0002-5. |
| [6] |
A. Constantin and J. Escher,
Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 303-328.
|
| [7] |
A. Constantin and J. Escher,
Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 51 (1998), 475-504.
|
| [8] |
A. Constantin and J. Escher,
Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.
doi: 10.1007/BF02392586. |
| [9] |
A. Constantin and J. Escher,
Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568.
doi: 10.4007/annals.2011.173.1.12. |
| [10] |
A. Constantin and W. A. Strauss,
Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610.
|
| [11] |
H.-H. Dai,
Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech., 127 (1998), 193-207.
doi: 10.1007/BF01170373. |
| [12] |
R. Danchin,
A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988.
|
| [13] |
H. R. Dullin, G. A. Gottwald and D. D. Holm, An integrable shallow water equation with linear and nonlinear dispersion, Phys. Rev. Lett., 87 (2001), 194501, 4501–4504.
doi: 10.1103/PhysRevLett.87.194501. |
| [14] |
A. Fokas,
On a class of physically important integrable equations, Phys. D, 87 (1995), 145-150.
doi: 10.1016/0167-2789(95)00133-O. |
| [15] |
G. Gui, Y. Liu, P. Olver and C. Qu,
Wave-breaking and peakons for a modified Camassa-Holm equation, Comm. Math. Phys., 319 (2013), 731-759.
doi: 10.1007/s00220-012-1566-0. |
| [16] |
A. A. Himonas and D. Mantzavinos,
Hölder continuity for the Fokas-Olver-Rosenau-Qiao equation, J. Nonlinear Sci., 24 (2014), 1105-1124.
doi: 10.1007/s00332-014-9212-y. |
| [17] |
X. Liu, Y. Liu and C. Qu,
Orbital stability of the train of peakons for an integrable modified Camassa-Holm equation, Adv. Math., 255 (2014), 1-37.
doi: 10.1016/j.aim.2013.12.032. |
| [18] |
Y. Liu and Z. Yin,
Global existence and blow-up phenomena for the Degasperis-Procesi equation, Comm. Math. Phys., 267 (2006), 801-820.
doi: 10.1007/s00220-006-0082-5. |
| [19] |
P. J. Olver and P. Rosenau,
Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53 (1996), 1900-1906.
doi: 10.1103/PhysRevE.53.1900. |
| [20] |
Z. Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons, J. Math. Phys., 47 (2006), 112701, 9 pp.
doi: 10.1063/1.2365758. |
| [21] |
C. Qu, X. Liu and Y. Liu,
Stability of peakons for an integrable modified Camassa-Holm equation with cubic nonlinearity, Comm. Math. Phys., 322 (2013), 967-997.
doi: 10.1007/s00220-013-1749-3. |
| [22] |
J. Song, C. Qu and Z. Qiao, A new integrable two-component system with cubic nonlinearity, J. Math. Phys., 52 (2011), 013503, 9 pp.
doi: 10.1063/1.3530865. |
| [23] |
G. B. Whitham, Linear and Nonlinear Waves, Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. |
| [24] |
Z. Xin and P. Zhang,
On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433.
|
| [25] |
K. Yan, Z. Qiao and Y. Zhang,
Blow-up phenomena for an integrable two-component Camassa-Holm system with cubic nonlinearity and peakon solutions, J. Differential Equations, 259 (2015), 6644-6671.
doi: 10.1016/j.jde.2015.08.004. |
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. D. Holm,
An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
| [3] |
R. M. Chen, F. Gao, Y. Liu and C. Qu,
Analysis on the blow-up of solutions to a class of integrable peakon equations, J. Funct. Anal., 270 (2016), 2343-2374.
doi: 10.1016/j.jfa.2016.01.017. |
| [4] |
A. Constantin,
Global existence of solutions and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.
doi: 10.5802/aif.1757. |
| [5] |
A. Constantin,
The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.
doi: 10.1007/s00222-006-0002-5. |
| [6] |
A. Constantin and J. Escher,
Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 303-328.
|
| [7] |
A. Constantin and J. Escher,
Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 51 (1998), 475-504.
|
| [8] |
A. Constantin and J. Escher,
Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.
doi: 10.1007/BF02392586. |
| [9] |
A. Constantin and J. Escher,
Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568.
doi: 10.4007/annals.2011.173.1.12. |
| [10] |
A. Constantin and W. A. Strauss,
Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610.
|
| [11] |
H.-H. Dai,
Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech., 127 (1998), 193-207.
doi: 10.1007/BF01170373. |
| [12] |
R. Danchin,
A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988.
|
| [13] |
H. R. Dullin, G. A. Gottwald and D. D. Holm, An integrable shallow water equation with linear and nonlinear dispersion, Phys. Rev. Lett., 87 (2001), 194501, 4501–4504.
doi: 10.1103/PhysRevLett.87.194501. |
| [14] |
A. Fokas,
On a class of physically important integrable equations, Phys. D, 87 (1995), 145-150.
doi: 10.1016/0167-2789(95)00133-O. |
| [15] |
G. Gui, Y. Liu, P. Olver and C. Qu,
Wave-breaking and peakons for a modified Camassa-Holm equation, Comm. Math. Phys., 319 (2013), 731-759.
doi: 10.1007/s00220-012-1566-0. |
| [16] |
A. A. Himonas and D. Mantzavinos,
Hölder continuity for the Fokas-Olver-Rosenau-Qiao equation, J. Nonlinear Sci., 24 (2014), 1105-1124.
doi: 10.1007/s00332-014-9212-y. |
| [17] |
X. Liu, Y. Liu and C. Qu,
Orbital stability of the train of peakons for an integrable modified Camassa-Holm equation, Adv. Math., 255 (2014), 1-37.
doi: 10.1016/j.aim.2013.12.032. |
| [18] |
Y. Liu and Z. Yin,
Global existence and blow-up phenomena for the Degasperis-Procesi equation, Comm. Math. Phys., 267 (2006), 801-820.
doi: 10.1007/s00220-006-0082-5. |
| [19] |
P. J. Olver and P. Rosenau,
Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53 (1996), 1900-1906.
doi: 10.1103/PhysRevE.53.1900. |
| [20] |
Z. Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons, J. Math. Phys., 47 (2006), 112701, 9 pp.
doi: 10.1063/1.2365758. |
| [21] |
C. Qu, X. Liu and Y. Liu,
Stability of peakons for an integrable modified Camassa-Holm equation with cubic nonlinearity, Comm. Math. Phys., 322 (2013), 967-997.
doi: 10.1007/s00220-013-1749-3. |
| [22] |
J. Song, C. Qu and Z. Qiao, A new integrable two-component system with cubic nonlinearity, J. Math. Phys., 52 (2011), 013503, 9 pp.
doi: 10.1063/1.3530865. |
| [23] |
G. B. Whitham, Linear and Nonlinear Waves, Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. |
| [24] |
Z. Xin and P. Zhang,
On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433.
|
| [25] |
K. Yan, Z. Qiao and Y. Zhang,
Blow-up phenomena for an integrable two-component Camassa-Holm system with cubic nonlinearity and peakon solutions, J. Differential Equations, 259 (2015), 6644-6671.
doi: 10.1016/j.jde.2015.08.004. |
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