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. 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.

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

[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.

[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.

[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. 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.

[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.

[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.

[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.

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.

[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. 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.

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

[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.

[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.

[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. 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.

[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.

[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.

[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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