# American Institute of Mathematical Sciences

June  2018, 38(6): 3085-3097. doi: 10.3934/dcds.2018134

## Liouville theorems for periodic two-component shallow water systems

 1 Department of Mathematics, South China Agricultural University, 510642 Guangzhou, China 2 Department of Mathematics, DePauw University, 46135 Greencastle, IN, USA 3 Department of Mathematics, Shandong University of Science and Technology, 266590 Qingdao, Shandong, China

Received  October 2017 Revised  January 2018 Published  April 2018

We establish Liouville-type theorems for periodic two-component shallow water systems, including a two-component Camassa-Holm equation (2CH) and a two-component Degasperis-Procesi (2DP) equation. More presicely, we prove that the only global, strong, spatially periodic solutions to the equations, vanishing at some point $(t_0, x_0)$, are the identically zero solutions. Also, we derive new local-in-space blow-up criteria for the dispersive 2CH and 2DP.

Citation: Qiaoyi Hu, Zhixin Wu, Yumei Sun. Liouville theorems for periodic two-component shallow water systems. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 3085-3097. doi: 10.3934/dcds.2018134
##### References:
 [1] L. Brandolese, Local-in-space criteria for blowup in shallow water and dispersive rod equations, Commun. Math. Phys., 330 (2014), 401-444.  doi: 10.1007/s00220-014-1958-4. [2] L. Brandolese and M. F. Cortez, Blowup issues for a class of nonlinear dispersive wave equations, J. Differential Equations, 256 (2014), 3981-3998.  doi: 10.1016/j.jde.2014.03.008. [3] L. Brandolese, A Liouville theorem for the Degasperis-Procesi equation, Ann. Sc. Norm. Super. Pisa Cl. Sci., 16 (2016), 759-765. [4] 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. [5] A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27.  doi: 10.1142/S0219530507000857. [6] R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661. [7] 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. [8] G. M. Coclite and K. H. Karlsen, On the well-posedness of the Degasperis-Procesi equation, J. Func. Anal., 233 (2006), 60-91.  doi: 10.1016/j.jfa.2005.07.008. [9] A. Constantin, The Cauchy problem for the periodic Camassa-Holm equation, J. Differential Equations, 141 (1997), 218-235.  doi: 10.1006/jdeq.1997.3333. [10] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.  doi: 10.1007/BF02392586. [11] 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. [12] A. Constantin and J. Escher, On the structure of a family of quasilinear equations arising in a shallow water theory, Math. Ann., 312 (1998), 403-416.  doi: 10.1007/s002080050228. [13] 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. [14] A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804.  doi: 10.1007/s00014-003-0785-6. [15] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.  doi: 10.1007/s00205-008-0128-2. [16] 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. [17] A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61.  doi: 10.1007/s002200050801. [18] A. Constantin and W. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610.  doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. [19] A. Degasperis and M. Procesi, Asymptotic integrability, Symmetry and Perurbation Theory (Rome, 1998), World Sci. Publishing, River Edge, NJ, 1999, 23–37. [20] A. Degasperis, D. D. Holm and A. N. W. Hone, A new integral equation with peakon solutions, Theo. Math. Phys., 133 (2002), 1463-1474. [21] 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), 194501-194504.  doi: 10.1103/PhysRevLett.87.194501. [22] 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. [23] 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. [24] 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. [25] J. Escher, Y. Liu and Z. Yin, Global weak solutions and blow-up structure for the Degasperis-Procesi equation, J. Func. Anal., 241 (2006), 457-485.  doi: 10.1016/j.jfa.2006.03.022. [26] 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. [27] 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. [28] D. T. Hoang, The local criteria for blowup of the Dullin-Gottwald-Holm equation and the two-component Dullin-Gottwald-Holm system, Ann. Fac. Sci. Toulouse, 25 (2016), 995-1012.  doi: 10.5802/afst.1519. [29] H. Holden and X. Raynaud, Global conservative solutions of the Camassa-Holm equation-a Lagrangian point of view, Comm. Partial Differential Equations, 32 (2007), 1511-1549.  doi: 10.1080/03605300601088674. [30] H. Holden and X. Raynaud, Dissipative solutions for the Camassa-Holm equation, Discrete Contin. Dyn. Syst., 24 (2009), 1047-1112.  doi: 10.3934/dcds.2009.24.1047. [31] Q. Hu and Z. Yin, Well-posedness and blowup phenomena for the periodic 2-component Camassa-Holm equation, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 93-107.  doi: 10.1017/S0308210509001218. [32] Q. Hu and Z. Yin, Blowup phenomena for a new periodic nonlinearly dispersive wave equation, Monatsh. Math., 165 (2012), 217-235.  doi: 10.1007/s00605-011-0293-5. [33] R. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82.  doi: 10.1017/S0022112001007224. [34] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in: Spectral Theory and Differential Equations, Lecture Notes in Math., Springer Verlag, Berlin, 448 (1975), 25–70. [35] J. Lenells, Traveling wave solutions of the Degasperis-Procesi equation, J. Math. Anal. Appl., 306 (2005), 72-82.  doi: 10.1016/j.jmaa.2004.11.038. [36] Y. Li and P. Oliver, Well-posedness and blow-up solutions for an integrable nonlinear dispersive model wave equation, J. Differential Equations, 162 (2000), 27-63.  doi: 10.1006/jdeq.1999.3683. [37] 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. [38] H. Lundmark, Formation and dynamics of shock waves in the Degasperis-Procesi equation, J. Nonlinear Sci., 17 (2007), 169-198.  doi: 10.1007/s00332-006-0803-3. [39] H. Lundmark and J. Szmigielski, Multi-peakon solutions of the Degasperis-Procesi equation, Inv. Prob., 19 (2003), 1241-1245.  doi: 10.1088/0266-5611/19/6/001. [40] G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208.  doi: 10.1016/S0393-0440(97)00010-7. [41] Z. Popowicz, A two-component generalization of the Degasperis-Procesi equation, J. Phys. A: Math. Gen., 39 (2006), 13717-13726.  doi: 10.1088/0305-4470/39/44/007. [42] G. Rodriguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46 (2001), 309-327.  doi: 10.1016/S0362-546X(01)00791-X. [43] V. Vakhnenko and E. Parkes, Periodic and solitary-wave solutions of the Degasperis-Procesi equation, Chaos, Solitons and Fractals, 20 (2004), 1059-1073.  doi: 10.1016/j.chaos.2003.09.043. [44] Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433.  doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5. [45] K. Yan and Z. Yin, On the Cauchy problem for a two-component Degasperis-Procesi system, J. Differential Equations, 252 (2012), 2131-2159.  doi: 10.1016/j.jde.2011.08.003. [46] Z. Yin, Global existence for a new periodic integrable equation, J. Math. Anal. Appl., 283 (2003), 129-139.  doi: 10.1016/S0022-247X(03)00250-6. [47] Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math., 47 (2003), 649-666. [48] Z. Yin, Global solutions to a new integrable equation with peakons, Indiana Univ. Math. J., 53 (2004), 1189-1209.  doi: 10.1512/iumj.2004.53.2479. [49] Z. Yin, Global weak solutions to a new periodic integrable equation with peakon solutions, J. Func. Anal., 212 (2004), 182-194.  doi: 10.1016/j.jfa.2003.07.010.

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##### References:
 [1] L. Brandolese, Local-in-space criteria for blowup in shallow water and dispersive rod equations, Commun. Math. Phys., 330 (2014), 401-444.  doi: 10.1007/s00220-014-1958-4. [2] L. Brandolese and M. F. Cortez, Blowup issues for a class of nonlinear dispersive wave equations, J. Differential Equations, 256 (2014), 3981-3998.  doi: 10.1016/j.jde.2014.03.008. [3] L. Brandolese, A Liouville theorem for the Degasperis-Procesi equation, Ann. Sc. Norm. Super. Pisa Cl. Sci., 16 (2016), 759-765. [4] 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. [5] A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27.  doi: 10.1142/S0219530507000857. [6] R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661. [7] 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. [8] G. M. Coclite and K. H. Karlsen, On the well-posedness of the Degasperis-Procesi equation, J. Func. Anal., 233 (2006), 60-91.  doi: 10.1016/j.jfa.2005.07.008. [9] A. Constantin, The Cauchy problem for the periodic Camassa-Holm equation, J. Differential Equations, 141 (1997), 218-235.  doi: 10.1006/jdeq.1997.3333. [10] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.  doi: 10.1007/BF02392586. [11] 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. [12] A. Constantin and J. Escher, On the structure of a family of quasilinear equations arising in a shallow water theory, Math. Ann., 312 (1998), 403-416.  doi: 10.1007/s002080050228. [13] 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. [14] A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804.  doi: 10.1007/s00014-003-0785-6. [15] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.  doi: 10.1007/s00205-008-0128-2. [16] 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. [17] A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61.  doi: 10.1007/s002200050801. [18] A. Constantin and W. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610.  doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. [19] A. Degasperis and M. Procesi, Asymptotic integrability, Symmetry and Perurbation Theory (Rome, 1998), World Sci. Publishing, River Edge, NJ, 1999, 23–37. [20] A. Degasperis, D. D. Holm and A. N. W. Hone, A new integral equation with peakon solutions, Theo. Math. Phys., 133 (2002), 1463-1474. [21] 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), 194501-194504.  doi: 10.1103/PhysRevLett.87.194501. [22] 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. [23] 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. [24] 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. [25] J. Escher, Y. Liu and Z. Yin, Global weak solutions and blow-up structure for the Degasperis-Procesi equation, J. Func. Anal., 241 (2006), 457-485.  doi: 10.1016/j.jfa.2006.03.022. [26] 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. [27] 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. [28] D. T. Hoang, The local criteria for blowup of the Dullin-Gottwald-Holm equation and the two-component Dullin-Gottwald-Holm system, Ann. Fac. Sci. Toulouse, 25 (2016), 995-1012.  doi: 10.5802/afst.1519. [29] H. Holden and X. Raynaud, Global conservative solutions of the Camassa-Holm equation-a Lagrangian point of view, Comm. Partial Differential Equations, 32 (2007), 1511-1549.  doi: 10.1080/03605300601088674. [30] H. Holden and X. Raynaud, Dissipative solutions for the Camassa-Holm equation, Discrete Contin. Dyn. Syst., 24 (2009), 1047-1112.  doi: 10.3934/dcds.2009.24.1047. [31] Q. Hu and Z. Yin, Well-posedness and blowup phenomena for the periodic 2-component Camassa-Holm equation, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 93-107.  doi: 10.1017/S0308210509001218. [32] Q. Hu and Z. Yin, Blowup phenomena for a new periodic nonlinearly dispersive wave equation, Monatsh. Math., 165 (2012), 217-235.  doi: 10.1007/s00605-011-0293-5. [33] R. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82.  doi: 10.1017/S0022112001007224. [34] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in: Spectral Theory and Differential Equations, Lecture Notes in Math., Springer Verlag, Berlin, 448 (1975), 25–70. [35] J. Lenells, Traveling wave solutions of the Degasperis-Procesi equation, J. Math. Anal. Appl., 306 (2005), 72-82.  doi: 10.1016/j.jmaa.2004.11.038. [36] Y. Li and P. Oliver, Well-posedness and blow-up solutions for an integrable nonlinear dispersive model wave equation, J. Differential Equations, 162 (2000), 27-63.  doi: 10.1006/jdeq.1999.3683. [37] 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. [38] H. Lundmark, Formation and dynamics of shock waves in the Degasperis-Procesi equation, J. Nonlinear Sci., 17 (2007), 169-198.  doi: 10.1007/s00332-006-0803-3. [39] H. Lundmark and J. Szmigielski, Multi-peakon solutions of the Degasperis-Procesi equation, Inv. Prob., 19 (2003), 1241-1245.  doi: 10.1088/0266-5611/19/6/001. [40] G. Misiolek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208.  doi: 10.1016/S0393-0440(97)00010-7. [41] Z. Popowicz, A two-component generalization of the Degasperis-Procesi equation, J. Phys. A: Math. Gen., 39 (2006), 13717-13726.  doi: 10.1088/0305-4470/39/44/007. [42] G. Rodriguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46 (2001), 309-327.  doi: 10.1016/S0362-546X(01)00791-X. [43] V. Vakhnenko and E. Parkes, Periodic and solitary-wave solutions of the Degasperis-Procesi equation, Chaos, Solitons and Fractals, 20 (2004), 1059-1073.  doi: 10.1016/j.chaos.2003.09.043. [44] Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433.  doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5. [45] K. Yan and Z. Yin, On the Cauchy problem for a two-component Degasperis-Procesi system, J. Differential Equations, 252 (2012), 2131-2159.  doi: 10.1016/j.jde.2011.08.003. [46] Z. Yin, Global existence for a new periodic integrable equation, J. Math. Anal. Appl., 283 (2003), 129-139.  doi: 10.1016/S0022-247X(03)00250-6. [47] Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math., 47 (2003), 649-666. [48] Z. Yin, Global solutions to a new integrable equation with peakons, Indiana Univ. Math. J., 53 (2004), 1189-1209.  doi: 10.1512/iumj.2004.53.2479. [49] Z. Yin, Global weak solutions to a new periodic integrable equation with peakon solutions, J. Func. Anal., 212 (2004), 182-194.  doi: 10.1016/j.jfa.2003.07.010.
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