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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[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.
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[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.
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[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.
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[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.
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[46]	G. Gui, Y. Liu and L. Tian, Global existence and blow-up phenomena for the peakon b-family of equations, Indiana Univ. Math. J., 57 (2008), 1209-1234. doi: 10.1512/iumj.2008.57.3213.
[47]	D. Henry, Compactly supported solutions of a family of nonlinear partial differential equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 15 (2008), 145-150.
[48]	D. Henry, Persistence properties for a family of nonlinear partial differential equations, Nonlinear Anal., 70 (2009), 1565-1573. doi: 10.1016/j.na.2008.02.104.
[49]	A. Himonas, G. Misiolek, G. Ponce and Y. Zhou, Persistence properties and unique continuation of solutions of the Camassa-Holm equation, Comm. Math. Phys., 27 (2007), 511-522. doi: 10.1007/s00220-006-0172-4.
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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. doi: 10.1016/j.anihpc.2011.04.003.
[44]	C. Guan and Z. Yin, On the existence of global weak solutions to an integrable two-component Camassa-Holm shallow-water system, Proc. Edinb. Math. Sic., 56 (2013), 755-775. doi: 10.1017/S0013091513000394.
[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.
[46]	G. Gui, Y. Liu and L. Tian, Global existence and blow-up phenomena for the peakon b-family of equations, Indiana Univ. Math. J., 57 (2008), 1209-1234. doi: 10.1512/iumj.2008.57.3213.
[47]	D. Henry, Compactly supported solutions of a family of nonlinear partial differential equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 15 (2008), 145-150.
[48]	D. Henry, Persistence properties for a family of nonlinear partial differential equations, Nonlinear Anal., 70 (2009), 1565-1573. doi: 10.1016/j.na.2008.02.104.
[49]	A. Himonas, G. Misiolek, G. Ponce and Y. Zhou, Persistence properties and unique continuation of solutions of the Camassa-Holm equation, Comm. Math. Phys., 27 (2007), 511-522. doi: 10.1007/s00220-006-0172-4.
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