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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

[3]

L. Brandolese, A Liouville theorem for the Degasperis-Procesi equation, Ann. Sc. Norm. Super. Pisa Cl. Sci., 16 (2016), 759-765.   Google Scholar

[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.  Google Scholar

[5]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27.  doi: 10.1142/S0219530507000857.  Google Scholar

[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.  Google Scholar

[7]

G. M. CocliteH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.  doi: 10.1007/BF02392586.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

A. Degasperis and M. Procesi, Asymptotic integrability, Symmetry and Perurbation Theory (Rome, 1998), World Sci. Publishing, River Edge, NJ, 1999, 23–37.  Google Scholar

[20]

A. DegasperisD. D. Holm and A. N. W. Hone, A new integral equation with peakon solutions, Theo. Math. Phys., 133 (2002), 1463-1474.   Google Scholar

[21]

H. R. DullinG. 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.  Google Scholar

[22]

J. EscherM. 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

[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.  Google Scholar

[24]

J. EscherO. 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

[25]

J. EscherY. 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.  Google Scholar

[26]

J. EscherY. 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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[47]

Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math., 47 (2003), 649-666.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[3]

L. Brandolese, A Liouville theorem for the Degasperis-Procesi equation, Ann. Sc. Norm. Super. Pisa Cl. Sci., 16 (2016), 759-765.   Google Scholar

[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.  Google Scholar

[5]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27.  doi: 10.1142/S0219530507000857.  Google Scholar

[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.  Google Scholar

[7]

G. M. CocliteH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.  doi: 10.1007/BF02392586.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

A. Degasperis and M. Procesi, Asymptotic integrability, Symmetry and Perurbation Theory (Rome, 1998), World Sci. Publishing, River Edge, NJ, 1999, 23–37.  Google Scholar

[20]

A. DegasperisD. D. Holm and A. N. W. Hone, A new integral equation with peakon solutions, Theo. Math. Phys., 133 (2002), 1463-1474.   Google Scholar

[21]

H. R. DullinG. 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.  Google Scholar

[22]

J. EscherM. 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

[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.  Google Scholar

[24]

J. EscherO. 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

[25]

J. EscherY. 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.  Google Scholar

[26]

J. EscherY. 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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[47]

Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math., 47 (2003), 649-666.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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