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Qualitative description of the particle trajectories for the N-solitons solution of the Korteweg-de Vries equation
On the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators
1. | Department of Mathematics, Sun Yat-sen University, Guangzhou 510275, China |
2. | Faculty of Information Technology, Macau University of Science and Technology, Macau, China |
In this paper, we mainly consider the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators: $m=(1-\partial_x^2)^su, s>1$. By Littlewood-Paley theory and transport equation theory, we first establish the local well-posedness of the generalized b-equation with fractional higher-order inertia operators which is the subsystem of the generalized two-component water wave system. Then we prove the local well-posedness of the generalized two-component water wave system with fractional higher-order inertia operators. Next, we present the blow-up criteria for these systems. Moreover, we obtain some global existence results for these systems.
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show all references
References:
[1] |
H. Bahouri, J. -Y. Chemin and R. Danchin,
Fourier Analysis and Nonlinear Partial Differential Equations Springer-Verlag Berlin Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[2] |
A. Bressan and A. Constantin,
Global conservative solutions of the Camassa-Holm equation, Arch. Rat. Mech. Anal., 183 (2007), 215-239.
doi: 10.1007/s00205-006-0010-z. |
[3] |
A. Bressan and A. Constantin,
Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27.
doi: 10.1142/S0219530507000857. |
[4] |
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. |
[5] |
R. Camassa and D. D. Holm,
A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33.
doi: 10.1016/S0065-2156(08)70254-0. |
[6] |
M. Chen, S.-Q. Liu and Y. Zhang,
A two-component generalization of the Camassa-Holm equation and its solutions, Lett. Math. Phys., 75 (2006), 1-15.
doi: 10.1007/s11005-005-0041-7. |
[7] |
G. M. Coclite and K. H. Karlsen,
On the well-posedness of the Degasperis-Procesi equation, J. Funct. Anal., 233 (2006), 60-91.
doi: 10.1016/j.jfa.2005.07.008. |
[8] |
G. M. Coclite, H. Holden and K. H. Karlsen,
Well-posedness of higher-order Camassa-Holm equations, J. Differential Equations, 246 (2009), 929-963.
doi: 10.1016/j.jde.2008.04.014. |
[9] |
A. Constantin,
On the inverse spectral problem for the Camassa-Holm equation, J. Funct. Anal., 155 (1998), 352-363.
doi: 10.1006/jfan.1997.3231. |
[10] |
A. Constantin,
Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.
doi: 10.5802/aif.1757. |
[11] |
A. Constantin,
On the scattering problem for the Camassa-Holm equation, Proc. Roy. Soc. London A, 457 (2001), 953-970.
doi: 10.1098/rspa.2000.0701. |
[12] |
A. Constantin, Finite propagation speed for the Camassa-Holm equation J. Math. Phys. , 46 (2005), 023506, 4pp.
doi: 10.1063/1.1845603. |
[13] |
A. Constantin and J. Escher,
Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.
doi: 10.1007/BF02392586. |
[14] |
A. Constantin and J. Escher,
Global existence and blow-up for a shallow water equation, Annali Sc. Norm. Sup. Pisa, 26 (1998), 303-328.
|
[15] |
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.
doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. |
[16] |
A. Constantin and J. Escher,
Global weak solutions for a shallow water equation, Indiana Uni. Math. J., 47 (1998), 1527-1545.
doi: 10.1512/iumj.1998.47.1466. |
[17] |
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. |
[18] |
A. Constantin, V. S. Gerdjikov and R. Ivanov,
Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207.
doi: 10.1088/0266-5611/22/6/017. |
[19] |
A. Constantin and R. Ivanov,
On an integrable two-component Camassa-Holm shallow water system, Phys. Lett. A, 372 (2008), 7129-7132.
doi: 10.1016/j.physleta.2008.10.050. |
[20] |
A. Constantin, R. Ivanov and J. Lenells,
Inverse scattering transform for the Degasperis-Procesi equation, Nonlinearity, 23 (2010), 2559-2575.
doi: 10.1088/0951-7715/23/10/012. |
[21] |
A. Constantin and B. Kolev,
Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helvetici, 78 (2003), 787-804.
doi: 10.1007/s00014-003-0785-6. |
[22] |
A. Constantin, T. Kappeler, B. Kolev and B. Topalov,
On geodesic exponential maps of the Virasoro group, Ann. Glob. Anal. Geom., 31 (2007), 155-180.
doi: 10.1007/s10455-006-9042-8. |
[23] |
A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982.
doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D. |
[24] |
A. Constantin and L. Molinet,
Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61.
doi: 10.1007/s002200050801. |
[25] |
R. Danchin,
Fourier Analysis Methods for PDEs, Lecture Notes, 14 November (2003). |
[26] |
R. Danchin,
A few remarks on the Camassa-Holm equation, Differential and Integral Equations, 14 (2001), 953-988.
|
[27] |
A. Degasperis and M. Procesi, Asymptotic integrability, Symmetry and perturbation theory (Rome 1998), pp. 23-37. World Sci. Publ. , River Edge, NJ, 1999. |
[28] |
J. Escher,
Non-metric two-component Euler equations on the circle, Monatsh. Math., 167 (2012), 449-459.
doi: 10.1007/s00605-011-0323-3. |
[29] |
J. Escher, D. Henry, B. Kolev and T. Lyons,
Two-component equations modelling water waves with constant vorticity, Ann. Mat. Pura Appl. (4), 195 (2016), 249-271.
doi: 10.1007/s10231-014-0461-z. |
[30] |
J. Escher and B. Kolev,
The Degasperis-Procesi equation as a non-metric Euler equation, Math. Z., 269 (2011), 1137-1153.
doi: 10.1007/s00209-010-0778-2. |
[31] |
J. Escher, M. Kohlmann and J. Lenells,
The geometry of the two-component Camassa-Holm and Degasperis-Procesi equations, J. Geom. Phys., 61 (2011), 436-452.
doi: 10.1016/j.geomphys.2010.10.011. |
[32] |
J. Escher, O. Lechtenfeld and Z. Yin,
Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation, Discrete Contin. Dyn. Syst., 19 (2007), 493-513.
doi: 10.3934/dcds.2007.19.493. |
[33] |
J. Escher, Y. Liu and Z. Yin,
Global weak solutions and blow-up structure for the Degasperis-Procesi equation, J. Funct. Anal., 241 (2006), 457-485.
doi: 10.1016/j.jfa.2006.03.022. |
[34] |
J. Escher, Y. Liu and Z. Yin,
Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation, Indiana Univ. Math. J., 56 (2007), 87-117.
doi: 10.1512/iumj.2007.56.3040. |
[35] |
J. Escher and T. Lyons,
Two-component higher order Camassa-Holm systems with fractional inertia operator: A geometric approach, J. Geom. Mech., 7 (2015), 281-293.
doi: 10.3934/jgm.2015.7.281. |
[36] |
J. Escher and Z. Yin,
Well-posedness, blow-up phenomena, and global solutions for the b-equation, J. Reine Angew. Math., 624 (2008), 51-80.
doi: 10.1515/CRELLE.2008.080. |
[37] |
J. Escher and Z. Yin,
Initial boundary value problems of the Camassa-Holm equation, Commun. Partial Differential Equations, 33 (2008), 377-395.
doi: 10.1080/03605300701318872. |
[38] |
J. Escher and Z. Yin,
Initial boundary value problems for nonlinear dispersive wave equations, J. Funct. Anal., 256 (2009), 479-508.
doi: 10.1016/j.jfa.2008.07.010. |
[39] |
C. Guan, H. He and Z. Yin,
Well-posedness, blow-up phenomena and persistence properties for a two-component water wave system, Nonlinear Anal. Real World Appl., 25 (2015), 219-237.
doi: 10.1016/j.nonrwa.2015.04.001. |
[40] |
C. Guan, H. Karlsen and Z. Yin,
Well-posedness and blow-up phenomena for a modified two-component Camassa-Holm equation, Contemporary Mathematics, 526 (2010), 199-220.
doi: 10.1090/conm/526/10382. |
[41] |
C. Guan and Z. Yin,
Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system, J. Differential Equations, 248 (2010), 2003-2014.
doi: 10.1016/j.jde.2009.08.002. |
[42] |
C. Guan and Z. Yin,
Global weak solutions for a two-component Camassa-Holm shallow water system, J. Funct. Anal., 260 (2011), 1132-1154.
doi: 10.1016/j.jfa.2010.11.015. |
[43] |
C. Guan and Z. Yin,
Global weak solutions for a modified two-component Camassa-Holm equation, Ann. I. H. Poincare-AN, 28 (2011), 623-641.
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