August  2013, 33(8): 3407-3441. doi: 10.3934/dcds.2013.33.3407

On the Cauchy problem for the two-component Dullin-Gottwald-Holm system

1. 

School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China

2. 

Jiangsu Key Laboratory for NSLSCS and School of Mathematical Sciences, Nanjing Normal University, Nanjing 210046, China

3. 

Department of Mathematics, University of Texas at Arlington, Arlington, TX 76019-0408

Received  May 2012 Revised  October 2012 Published  January 2013

Considered herein is the initial-value problem for a two-component Dullin-Gottwald-Holm system. The local well-posedness in the Sobolev space $H^{s}(\mathbb{R})$ with $s>3/2$ is established by using the bi-linear estimate technique to the approximate solutions. Then the wave-breaking criteria and global solutions are determined in $H^s(\mathbb{R}), s > 3/2.$ Finally, existence of the solitary-wave solutions is demonstrated.
Citation: Yong Chen, Hongjun Gao, Yue Liu. On the Cauchy problem for the two-component Dullin-Gottwald-Holm system. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3407-3441. doi: 10.3934/dcds.2013.33.3407
References:
[1]

M. Ben-Artzi, H. Koch and J. C. Saut, Dispersion estimates for third order equations in two dimensions, Comm. Partial Differential Equations, 28 (2003), 1943-1974. doi: 10.1081/PDE-120025491.

[2]

J. L. Bona and Y. A. Li, Decay and analyticity of solitary waves, J. Math. Pures Appl., 76 (1997), 377-430. doi: 10.1016/S0021-7824(97)89957-6.

[3]

J. L. Bona and R. Smith, The initial value problem for the Korteweg-de Vries equation, Phil. Trans. Roy. Soc. London A, 278 (1975), 555-601.

[4]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, I, I. Schrödinger equations, Geom. Funct. Anal., 2 (1993), 107-156. doi: 10.1007/BF01896020.

[5]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, II,. The KdV-equation, Geom. Funct. Anal., 3 (1993), 209-262. doi: 10.1007/BF01895688.

[6]

P. J. Byers, "The Initial Value Problem for a KdV-Type Equation and Related Bilinear Estimate," Ph.D thesis, University of Notre Dame, 2003.

[7]

J. C. Burns, Long waves on running water, Math. Proc. Cambridge Philos. Soc., 49 (1953), 695-706.

[8]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness for the KdV in Sobolev spaces of negative index, Electron. J. Differential Equations, 26 (2001), 1-7.

[9]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbfR$ and $\mathbfT$, J. Amer. Math. Soc., 16 (2003), 705-749. doi: 10.1090/S0894-0347-03-00421-1.

[10]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness for Schrödinger equations with derivative, SIAM J. Math. Anal., 33 (2001), 649-666. doi: 10.1137/S0036141001384387.

[11]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.

[12]

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

[13]

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.

[14]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa, 26 (1998), 303-328.

[15]

A. Constantin and R. Ivanov, On the integrable two-component Camassa-Holm shallow water system, Phys. Lett. A, 372 (2008), 7129-7132. doi: 10.1016/j.physleta.2008.10.050.

[16]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math.,57 (2004), 481-527. doi: 10.1002/cpa.3046.

[17]

A. Constantin and W. A. Strauss, Stability of the Camassa-Holm solitons, J. Nonlinear Sci., 12 (2002), 415-22. doi: 10.1007/s00332-002-0517-x.

[18]

A. de Bouard, A. Debussche and Y. Tsutsumi, White noise driven Korteweg-de Vries equation, J. Funct. Anal., 169 (1999), 532-558. doi: 10.1006/jfan.1999.3484.

[19]

H. Dullin, G. Gottwald and D. Holm, An integrable shallow water equation with linear and nonlinear dispersion, Phys. Rev. Lett., 87 (2001), 4501-4504.

[20]

J. Escher, O. Lechtenfeld and Z. Y. Yin, Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation, Discrete Contin. Dynam. Systems, 19 (2007), 493-513. doi: 10.3934/dcds.2007.19.493.

[21]

Y. Fu, Y. Liu and C. Z. Qu, Well-posedness and blow-up solution for a modified two-component periodic Camassa-Holm system with peakons, Math. Ann., 348 (2010), 415-448. doi: 10.1007/s00208-010-0483-9.

[22]

C. X. Guan and Z. Y. Yin, Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water systems, J. Differential Equations, 248 (2010), 2003-2014. doi: 10.1016/j.jde.2009.08.002.

[23]

C. X. Guan and Z. Y. Yin, Global weak solutions for a two-component Camassa-Holm shallow water systems, J. Funct. Anal., 260 (2011), 1132-1154. doi: 10.1016/j.jfa.2010.11.015.

[24]

C. X. Guan and Z. Y. 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.

[25]

G. L. Gui and Y. Liu, On the global existence and wave-breaking criteria for the two-component Camassa-Holm system, J. Funct. Anal., 258 (2010), 4251-4278. doi: 10.1016/j.jfa.2010.02.008.

[26]

G. L. Gui and Y. Liu, On the Cauchy problem for the two-component Camassa-Holm system, Math. Z., 268 (2011), 45-66. doi: 10.1007/s00209-009-0660-2.

[27]

F. Guo, H. J. Gao and Y. Liu, Blow-up mechanism and global solutions for the two-component Dullin-Gottwald-Holm system,, Accepted by J. of the London Math. Soc., (). 

[28]

A. Himonas and G. Misiolek, The Cauchy problem for a shallow water type equation, Comm. Partial Differential Equations, 23 (1998), 123-139. doi: 10.1080/03605309808821340.

[29]

A. Himonas and G. Misiolek, Well-posedness of the Cauchy problem for a shallow water equation on the circle, J. Differential Equations, 161 (2000), 479-495. doi: 10.1006/jdeq.1999.3695.

[30]

A. Himonas and G. Misiolek, The initial value problem for a fifth order shallow water, in "Analysis, Geometry, Number Theory: The Mathematics of Leon Ehrenpreis Contemp. Math." 251, Amer. Math. Soc., Providence, RI, (2000), 309-320. doi: 10.1090/conm/251/03878.

[31]

R. Ivanov, Two-component integrable systems modelling shallow water waves: The constant vorticity case, Wave Motion, 46 (2009), 389-396. doi: 10.1016/j.wavemoti.2009.06.012.

[32]

Y. L. Jia and Z. H. Huo, Well-posedness for the fifth-order shallow water equations, J. Differential Equations, 246 (2009), 2448-2467. doi: 10.1016/j.jde.2008.10.027.

[33]

R. S. Johnson, The Camassa-Holm equation for water waves moving over a shear flow, Fluid Dynam. Res., 33 (2003), 97-111. doi: 10.1016/S0169-5983(03)00036-4.

[34]

T. Kato, On the Korteweg-de Vries equation, Manuscripta Math., 28 (1979), 89-99.

[35]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equation, Commun. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704.

[36]

C. E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 40 (1991), 33-69. doi: 10.1512/iumj.1991.40.40003.

[37]

C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21. doi: 10.1215/S0012-7094-93-07101-3.

[38]

C. E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603. doi: 10.1090/S0894-0347-96-00200-7.

[39]

Y. A. Li and P. J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162 (2000), 27-63. doi: 10.1006/jdeq.1999.3683.

[40]

X. Liu and Y. Jin, The Cauchy problem of a shallow water equation, Acta Math. Sin. (Engl. Ser.), 30 (2004), 1-16. doi: 10.1007/s10114-004-0420-5.

[41]

L. Molinet, J. C. Saut and N. Tzvetkov, Global well-posedness for the KP-I equation, Math. Ann., 324 (2002), 255-275. doi: 10.1007/s00208-002-0338-0.

[42]

P. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E., 53 (1996), 1900-1906. doi: 10.1103/PhysRevE.53.1900.

[43]

T. Tao, Multilinear weighted convolution of $L^2$ functions, and applications to nonlinear dispersive equation, Amer J. Math., 123 (2001), 839-908.

[44]

L. X. Tian, G. L. Gui and Y. Liu, On the well-posedness problem and the scattering problem for the Dullin-Gottwald-Holm equation, Comm. Math. Phys., 257 (2005), 667-701. doi: 10.1007/s00220-005-1356-z.

[45]

H. Wang and S. B. Cui, Global well-posedness of the Cauchy problem of the fifth-order shallow water equation, J. Differential Equations, 230 (2006), 600-613. doi: 10.1016/j.jde.2006.04.008.

[46]

X. Y. Yang and Y. S. Li, Global well-posedness for a fifth-order shallow water equation in Sobolev spaces, J. Differential Equations, 248 (2010), 1458-1472. doi: 10.1016/j.jde.2010.01.004.

[47]

P. Z. Zhang and Y. Liu, Stability of solitary waves and wave-breaking phenomena for the two-component Camassa-Holm system, Int. Math. Res. Not. IMRN, 11 (2010), 1981-2021. doi: 10.1093/imrn/rnp211.

show all references

References:
[1]

M. Ben-Artzi, H. Koch and J. C. Saut, Dispersion estimates for third order equations in two dimensions, Comm. Partial Differential Equations, 28 (2003), 1943-1974. doi: 10.1081/PDE-120025491.

[2]

J. L. Bona and Y. A. Li, Decay and analyticity of solitary waves, J. Math. Pures Appl., 76 (1997), 377-430. doi: 10.1016/S0021-7824(97)89957-6.

[3]

J. L. Bona and R. Smith, The initial value problem for the Korteweg-de Vries equation, Phil. Trans. Roy. Soc. London A, 278 (1975), 555-601.

[4]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, I, I. Schrödinger equations, Geom. Funct. Anal., 2 (1993), 107-156. doi: 10.1007/BF01896020.

[5]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, II,. The KdV-equation, Geom. Funct. Anal., 3 (1993), 209-262. doi: 10.1007/BF01895688.

[6]

P. J. Byers, "The Initial Value Problem for a KdV-Type Equation and Related Bilinear Estimate," Ph.D thesis, University of Notre Dame, 2003.

[7]

J. C. Burns, Long waves on running water, Math. Proc. Cambridge Philos. Soc., 49 (1953), 695-706.

[8]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness for the KdV in Sobolev spaces of negative index, Electron. J. Differential Equations, 26 (2001), 1-7.

[9]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbfR$ and $\mathbfT$, J. Amer. Math. Soc., 16 (2003), 705-749. doi: 10.1090/S0894-0347-03-00421-1.

[10]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness for Schrödinger equations with derivative, SIAM J. Math. Anal., 33 (2001), 649-666. doi: 10.1137/S0036141001384387.

[11]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.

[12]

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

[13]

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.

[14]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa, 26 (1998), 303-328.

[15]

A. Constantin and R. Ivanov, On the integrable two-component Camassa-Holm shallow water system, Phys. Lett. A, 372 (2008), 7129-7132. doi: 10.1016/j.physleta.2008.10.050.

[16]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math.,57 (2004), 481-527. doi: 10.1002/cpa.3046.

[17]

A. Constantin and W. A. Strauss, Stability of the Camassa-Holm solitons, J. Nonlinear Sci., 12 (2002), 415-22. doi: 10.1007/s00332-002-0517-x.

[18]

A. de Bouard, A. Debussche and Y. Tsutsumi, White noise driven Korteweg-de Vries equation, J. Funct. Anal., 169 (1999), 532-558. doi: 10.1006/jfan.1999.3484.

[19]

H. Dullin, G. Gottwald and D. Holm, An integrable shallow water equation with linear and nonlinear dispersion, Phys. Rev. Lett., 87 (2001), 4501-4504.

[20]

J. Escher, O. Lechtenfeld and Z. Y. Yin, Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation, Discrete Contin. Dynam. Systems, 19 (2007), 493-513. doi: 10.3934/dcds.2007.19.493.

[21]

Y. Fu, Y. Liu and C. Z. Qu, Well-posedness and blow-up solution for a modified two-component periodic Camassa-Holm system with peakons, Math. Ann., 348 (2010), 415-448. doi: 10.1007/s00208-010-0483-9.

[22]

C. X. Guan and Z. Y. Yin, Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water systems, J. Differential Equations, 248 (2010), 2003-2014. doi: 10.1016/j.jde.2009.08.002.

[23]

C. X. Guan and Z. Y. Yin, Global weak solutions for a two-component Camassa-Holm shallow water systems, J. Funct. Anal., 260 (2011), 1132-1154. doi: 10.1016/j.jfa.2010.11.015.

[24]

C. X. Guan and Z. Y. 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.

[25]

G. L. Gui and Y. Liu, On the global existence and wave-breaking criteria for the two-component Camassa-Holm system, J. Funct. Anal., 258 (2010), 4251-4278. doi: 10.1016/j.jfa.2010.02.008.

[26]

G. L. Gui and Y. Liu, On the Cauchy problem for the two-component Camassa-Holm system, Math. Z., 268 (2011), 45-66. doi: 10.1007/s00209-009-0660-2.

[27]

F. Guo, H. J. Gao and Y. Liu, Blow-up mechanism and global solutions for the two-component Dullin-Gottwald-Holm system,, Accepted by J. of the London Math. Soc., (). 

[28]

A. Himonas and G. Misiolek, The Cauchy problem for a shallow water type equation, Comm. Partial Differential Equations, 23 (1998), 123-139. doi: 10.1080/03605309808821340.

[29]

A. Himonas and G. Misiolek, Well-posedness of the Cauchy problem for a shallow water equation on the circle, J. Differential Equations, 161 (2000), 479-495. doi: 10.1006/jdeq.1999.3695.

[30]

A. Himonas and G. Misiolek, The initial value problem for a fifth order shallow water, in "Analysis, Geometry, Number Theory: The Mathematics of Leon Ehrenpreis Contemp. Math." 251, Amer. Math. Soc., Providence, RI, (2000), 309-320. doi: 10.1090/conm/251/03878.

[31]

R. Ivanov, Two-component integrable systems modelling shallow water waves: The constant vorticity case, Wave Motion, 46 (2009), 389-396. doi: 10.1016/j.wavemoti.2009.06.012.

[32]

Y. L. Jia and Z. H. Huo, Well-posedness for the fifth-order shallow water equations, J. Differential Equations, 246 (2009), 2448-2467. doi: 10.1016/j.jde.2008.10.027.

[33]

R. S. Johnson, The Camassa-Holm equation for water waves moving over a shear flow, Fluid Dynam. Res., 33 (2003), 97-111. doi: 10.1016/S0169-5983(03)00036-4.

[34]

T. Kato, On the Korteweg-de Vries equation, Manuscripta Math., 28 (1979), 89-99.

[35]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equation, Commun. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704.

[36]

C. E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 40 (1991), 33-69. doi: 10.1512/iumj.1991.40.40003.

[37]

C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21. doi: 10.1215/S0012-7094-93-07101-3.

[38]

C. E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603. doi: 10.1090/S0894-0347-96-00200-7.

[39]

Y. A. Li and P. J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162 (2000), 27-63. doi: 10.1006/jdeq.1999.3683.

[40]

X. Liu and Y. Jin, The Cauchy problem of a shallow water equation, Acta Math. Sin. (Engl. Ser.), 30 (2004), 1-16. doi: 10.1007/s10114-004-0420-5.

[41]

L. Molinet, J. C. Saut and N. Tzvetkov, Global well-posedness for the KP-I equation, Math. Ann., 324 (2002), 255-275. doi: 10.1007/s00208-002-0338-0.

[42]

P. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E., 53 (1996), 1900-1906. doi: 10.1103/PhysRevE.53.1900.

[43]

T. Tao, Multilinear weighted convolution of $L^2$ functions, and applications to nonlinear dispersive equation, Amer J. Math., 123 (2001), 839-908.

[44]

L. X. Tian, G. L. Gui and Y. Liu, On the well-posedness problem and the scattering problem for the Dullin-Gottwald-Holm equation, Comm. Math. Phys., 257 (2005), 667-701. doi: 10.1007/s00220-005-1356-z.

[45]

H. Wang and S. B. Cui, Global well-posedness of the Cauchy problem of the fifth-order shallow water equation, J. Differential Equations, 230 (2006), 600-613. doi: 10.1016/j.jde.2006.04.008.

[46]

X. Y. Yang and Y. S. Li, Global well-posedness for a fifth-order shallow water equation in Sobolev spaces, J. Differential Equations, 248 (2010), 1458-1472. doi: 10.1016/j.jde.2010.01.004.

[47]

P. Z. Zhang and Y. Liu, Stability of solitary waves and wave-breaking phenomena for the two-component Camassa-Holm system, Int. Math. Res. Not. IMRN, 11 (2010), 1981-2021. doi: 10.1093/imrn/rnp211.

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