On the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators

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March 2017, 37(3): 1509-1537. doi: 10.3934/dcds.2017062

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

¹Corresponding author

Received June 2016 Revised September 2016 Published December 2016

Fund Project: This work was partially supported by NNSFC (No.11671407 and No.11271382), FDCT (No. 098/2013/A3), Guangdong Special Support Program (No. 8-2015), and the key project of NSF of Guangdong province (No. 2016A030311004)

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.

Keywords: Two-component shallow water wave system with fractional higher-order inertia operators, Littlewood-Paley theory, local well-posedness, blow-up criteria, global existence.

Mathematics Subject Classification: Primary:35G25;Secondary:35L05, 35Q53.

Citation: Huijun He, Zhaoyang Yin. On the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1509-1537. doi: 10.3934/dcds.2017062

References:

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References:

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[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. Google Scholar
[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. Google Scholar
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[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. Google Scholar
[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. Google Scholar
[25]	R. Danchin, Fourier Analysis Methods for PDEs, Lecture Notes, 14 November (2003).Google Scholar
[26]	R. Danchin, A few remarks on the Camassa-Holm equation, Differential and Integral Equations, 14 (2001), 953-988. Google Scholar
[27]	A. Degasperis and M. Procesi, Asymptotic integrability, Symmetry and perturbation theory (Rome 1998), pp. 23-37. World Sci. Publ. , River Edge, NJ, 1999. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
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[45]	C. Guan and Z. Yin, On the global weak solutions for a modified two-component Camassa-Holm equation, Math. Nach., 286 (2013), 1287-1304. doi: 10.1002/mana.201200193. Google Scholar
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