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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.

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 and Continuous Dynamical Systems, 2017, 37 (3) : 1509-1537. doi: 10.3934/dcds.2017062
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. 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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. 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