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 & Continuous Dynamical Systems - A, 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. doi: 10.1081/PDE-120025491. Google Scholar

[2]

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

[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. Google Scholar

[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. doi: 10.1007/BF01896020. Google Scholar

[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. doi: 10.1007/BF01895688. Google Scholar

[6]

P. J. Byers, "The Initial Value Problem for a KdV-Type Equation and Related Bilinear Estimate,", Ph.D thesis, (2003). Google Scholar

[7]

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

[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. Google Scholar

[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. doi: 10.1090/S0894-0347-03-00421-1. Google Scholar

[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. doi: 10.1137/S0036141001384387. Google Scholar

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

[12]

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

[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. doi: 10.1007/PL00004793. Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[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. doi: 10.3934/dcds.2007.19.493. Google Scholar

[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. doi: 10.1007/s00208-010-0483-9. Google Scholar

[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. doi: 10.1016/j.jde.2009.08.002. Google Scholar

[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. doi: 10.1016/j.jfa.2010.11.015. Google Scholar

[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. doi: 10.1016/j.anihpc.2011.04.003. Google Scholar

[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. doi: 10.1016/j.jfa.2010.02.008. Google Scholar

[26]

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

[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., (). Google Scholar

[28]

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

[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. doi: 10.1006/jdeq.1999.3695. Google Scholar

[30]

A. Himonas and G. Misiolek, The initial value problem for a fifth order shallow water,, in, 251 (2000), 309. doi: 10.1090/conm/251/03878. Google Scholar

[31]

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

[32]

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

[33]

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

[34]

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

[35]

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

[36]

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

[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. doi: 10.1215/S0012-7094-93-07101-3. Google Scholar

[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. doi: 10.1090/S0894-0347-96-00200-7. Google Scholar

[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. doi: 10.1006/jdeq.1999.3683. Google Scholar

[40]

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

[41]

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

[42]

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

[43]

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

[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. doi: 10.1007/s00220-005-1356-z. Google Scholar

[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. doi: 10.1016/j.jde.2006.04.008. Google Scholar

[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. doi: 10.1016/j.jde.2010.01.004. Google Scholar

[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. doi: 10.1093/imrn/rnp211. Google Scholar

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. doi: 10.1081/PDE-120025491. Google Scholar

[2]

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

[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. Google Scholar

[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. doi: 10.1007/BF01896020. Google Scholar

[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. doi: 10.1007/BF01895688. Google Scholar

[6]

P. J. Byers, "The Initial Value Problem for a KdV-Type Equation and Related Bilinear Estimate,", Ph.D thesis, (2003). Google Scholar

[7]

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

[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. Google Scholar

[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. doi: 10.1090/S0894-0347-03-00421-1. Google Scholar

[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. doi: 10.1137/S0036141001384387. Google Scholar

[11]

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

[12]

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

[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. doi: 10.1007/PL00004793. Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[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. doi: 10.3934/dcds.2007.19.493. Google Scholar

[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. doi: 10.1007/s00208-010-0483-9. Google Scholar

[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. doi: 10.1016/j.jde.2009.08.002. Google Scholar

[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. doi: 10.1016/j.jfa.2010.11.015. Google Scholar

[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. doi: 10.1016/j.anihpc.2011.04.003. Google Scholar

[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. doi: 10.1016/j.jfa.2010.02.008. Google Scholar

[26]

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

[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., (). Google Scholar

[28]

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

[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. doi: 10.1006/jdeq.1999.3695. Google Scholar

[30]

A. Himonas and G. Misiolek, The initial value problem for a fifth order shallow water,, in, 251 (2000), 309. doi: 10.1090/conm/251/03878. Google Scholar

[31]

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

[32]

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

[33]

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

[34]

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

[35]

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

[36]

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

[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. doi: 10.1215/S0012-7094-93-07101-3. Google Scholar

[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. doi: 10.1090/S0894-0347-96-00200-7. Google Scholar

[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. doi: 10.1006/jdeq.1999.3683. Google Scholar

[40]

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

[41]

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

[42]

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

[43]

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

[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. doi: 10.1007/s00220-005-1356-z. Google Scholar

[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. doi: 10.1016/j.jde.2006.04.008. Google Scholar

[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. doi: 10.1016/j.jde.2010.01.004. Google Scholar

[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. doi: 10.1093/imrn/rnp211. Google Scholar

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