# American Institute of Mathematical Sciences

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, 2013, 33 (8) : 3407-3441. doi: 10.3934/dcds.2013.33.3407
##### References:
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Anal., 33 (2001), 649-666. 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), 321-362.  Google Scholar [12] 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 [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.  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), 303-328.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  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), 4501-4504. 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), 493-513. doi: 10.3934/dcds.2007.19.493.  Google Scholar [21] Y. Fu, Y. Liu and C. Z. 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Anal., 258 (2010), 4251-4278. 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), 45-66. 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), 123-139. 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), 479-495. 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 "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.  Google Scholar [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.  Google Scholar [32] Y. L. Jia and Z. H. Huo, Well-posedness for the fifth-order shallow water equations, J. Differential Equations, 246 (2009), 2448-2467. 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), 97-111. doi: 10.1016/S0169-5983(03)00036-4.  Google Scholar [34] T. Kato, On the Korteweg-de Vries equation, Manuscripta Math., 28 (1979), 89-99. Google Scholar [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.  Google Scholar [36] C. E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 40 (1991), 33-69. 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), 1-21. 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), 573-603. 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), 27-63. 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), 1-16. 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), 255-275. 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), 1900-1906. 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), 839-908.  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), 667-701. 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), 600-613. 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), 1458-1472. 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), 1981-2021. 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), 1943-1974. 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), 377-430. 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), 555-601.  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), 107-156. 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), 209-262. 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, University of Notre Dame, 2003.  Google Scholar [7] J. C. Burns, Long waves on running water, Math. Proc. Cambridge Philos. Soc., 49 (1953), 695-706.  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), 1-7.  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), 705-749. 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), 649-666. 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), 321-362.  Google Scholar [12] 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 [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.  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), 303-328.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  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), 4501-4504. 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), 493-513. 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), 415-448. 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), 2003-2014. 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), 1132-1154. 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), 623-641. 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), 4251-4278. 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), 45-66. 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), 123-139. 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), 479-495. 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 "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.  Google Scholar [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.  Google Scholar [32] Y. L. Jia and Z. H. Huo, Well-posedness for the fifth-order shallow water equations, J. Differential Equations, 246 (2009), 2448-2467. 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), 97-111. doi: 10.1016/S0169-5983(03)00036-4.  Google Scholar [34] T. Kato, On the Korteweg-de Vries equation, Manuscripta Math., 28 (1979), 89-99. Google Scholar [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.  Google Scholar [36] C. E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 40 (1991), 33-69. 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), 1-21. 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), 573-603. 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), 27-63. 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), 1-16. 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), 255-275. 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), 1900-1906. 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), 839-908.  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), 667-701. 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), 600-613. 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), 1458-1472. 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), 1981-2021. doi: 10.1093/imrn/rnp211.  Google Scholar
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