# American Institute of Mathematical Sciences

April  2013, 33(4): 1699-1712. doi: 10.3934/dcds.2013.33.1699

## Well-posedness for a modified two-component Camassa-Holm system in critical spaces

 1 Department of Mathematics, Sun Yat-sen University, 510275 Guangzhou, China

Received  June 2011 Revised  September 2012 Published  October 2012

This paper is concerned with the problem of well-posedness for a modified two-component Camassa-Holm system in Besov spaces with the critical index $s=\frac 3 2$.
Citation: Kai Yan, Zhaoyang Yin. Well-posedness for a modified two-component Camassa-Holm system in critical spaces. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1699-1712. doi: 10.3934/dcds.2013.33.1699
##### References:
 [1] R. Beals, D. Scattinger and J. Szmigielski, Acoustic scatting and the extended Korteweg-de Vries hierarchy, Adv. Math., 140 (1998), 190-206. doi: 10.1006/aima.1998.1768.  Google Scholar [2] 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 [3] A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl. (Singap.), 5 (2007), 1-27.  Google Scholar [4] 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 [5] R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. doi: 10.1016/S0065-2156(08)70254-0.  Google Scholar [6] J.-Y. Chemin, "Perfect Incompressible Fluids," in "Oxford Lecture Series in Mathematics and its Applications", Vol. 14, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar [7] G. M. Coclite, H. Holden and K. H. Karlsen., Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM J. Math. Anal., 37 (2006), 1044-1069. doi: 10.1137/040616711.  Google Scholar [8] A. Constantin, The Hamiltonian structure of the Camassa-Holm equation, Exposition. Math., 15 (1997), 53-85.  Google Scholar [9] A. Constantin, Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362. doi: 10.5802/aif.1757.  Google Scholar [10] A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. Roy. Soc. London A, 457 (2001), 953-970. doi: 10.1098/rspa.2000.0701.  Google Scholar [11] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.  Google Scholar [12] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 1992 (2009), 165-186. doi: 10.1007/s00205-008-0128-2.  Google Scholar [13] A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Annali Sc. Norm. Sup. Pisa, 26 (1998), 303-328.  Google Scholar [14] 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 [15] 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 [16] A. Constantin and J. Escher, On the blow-up rate and the blow-up of breaking waves for a shallow water equation, Math. Z., 233 (2000), 75-91. doi: 10.1007/PL00004793.  Google Scholar [17] A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431. doi: 10.1090/S0273-0979-07-01159-7.  Google Scholar [18] A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12.  Google Scholar [19] 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 [20] 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>3.3.CO;2-C.  Google Scholar [21] H. Dai, Model equations for nolinear dispersive waves in compressible Mooney-Rivlin rod, Acta Mech., 127 (1998), 193-207. doi: 10.1007/BF01170373.  Google Scholar [22] R. Dachin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988.  Google Scholar [23] R. Dachin, A note on well-posedness for Camassa-Holm equation, J. Differential Equations, 192 (2003), 429-444. doi: 10.1016/S0022-0396(03)00096-2.  Google Scholar [24] R. Dachin, "Fourier Analysis Methods for PDEs," Lecture Notes, 14 November 2003. Google Scholar [25] H. R. Dullin, G. A. Gottwald and D. D. Holm, An integral shallow water equation with linear and nonlinear dispersion, Phys. Rev. Lett., 87 (2001), 4501-4504. doi: 10.1103/PhysRevLett.87.194501.  Google Scholar [26] J. Escher and Z. Yin, Initial boundary value problems of the Camassa-Holm equation, Comm. Partial Differential Equations, 33 (2008), 377-395. doi: 10.1080/03605300701318872.  Google Scholar [27] J. Escher and Z. Yin, Initial boundary value problems for nonlinear dispersive wave equations, J. Funct. Anal., 256 (2009), 479-508. doi: 10.1016/j.jfa.2008.07.010.  Google Scholar [28] A. Fokas and B. Fuchssteiner, Symplectic structures, their B$\ddota$cklund transformations and hereditary symmetries, Phys. D, 4 (1981), 47-66.  Google Scholar [29] C. Guan, K. H. Karlsen and Z. Yin, Well-posedness and blow-up phenomena for a modified two-component Camassa-Holm equation, Contemp. Math., 526 (2010), 199-220. doi: 10.1090/conm/526/10382.  Google Scholar [30] C. Guan and Z. Yin, Global weak solutions for a modified two-component Camassa-Holm equation, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 28 (2011), 623-641. doi: 10.1016/j.anihpc.2011.04.003.  Google Scholar [31] G. 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 [32] 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 [33] D. Holm, J. Marsden and T. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81.  Google Scholar [34] D. Holm, L. Naraigh and C. Tronci, Singular solution of a modified two-component Camassa-Holm equation, Phys. Rev. E(3),79 (2009), 1-13.  Google Scholar [35] D. Ionescu-Krus, Variational derivation of the Camassa-Holm shallow water equation, J. Nonlinear Math. Phys., 14 (2007), 303-312. doi: 10.2991/jnmp.2007.14.3.1.  Google Scholar [36] R. Ivanov, Water waves and integrability, Philos. Trans. R. Soc. Lond. Ser. A., Math. Phys. Eng. Sci., 365 (2007), 2267-2280. doi: 10.1098/rsta.2007.2007.  Google Scholar [37] R. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 457 (2002), 63-82.  Google Scholar [38] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in "Spectral Theory and Differential Equations", Lecture Notes in Math., 448, Springer Verlag, Berlin, (1975), 25-70.  Google Scholar [39] 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 [40] X. Liu and Z. Yin, On the low regularity solutions for a modified two-component Camassa-Holm shallow water system, Glasgow. Math. J., 53 (2011), 611-621. doi: 10.1017/S0017089511000176.  Google Scholar [41] J. Marsden and T. Ratiu, "Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanics and Systems," 2nd ed., Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1999.  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] W. Tan and Z. Yin, Global dissipative solutions of a modified two-component Camassa-Holm equation shallow water system, J. Math. Phys., 52 (2011), 1-24. 033507. doi: 10.1063/1.3562928.  Google Scholar [44] W. Tan and Z. Yin, Global periodic conservative solutions of a periodic modified two-component Camassa-Holm equation, J. Funct. Anal., 261 (2011), 1204-1226. doi: 10.1016/j.jfa.2011.04.015.  Google Scholar [45] J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.  Google Scholar [46] G. B. Whitham, "Linear and Nonlinear Waves," J. Wiley and Sons, New York, 1980.  Google Scholar [47] 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>3.3.CO;2-X.  Google Scholar [48] K. Yan and Z. Yin, Analytic solutions of the Cauchy problem for two-component shallow water systems, Math. Z., 269 (2011), 1113-1127. doi: 10.1007/s00209-010-0775-5.  Google Scholar [49] Z. Yin, Well-posedness, blowup, and global existence for an integrable shallow water equation, Discrete Contin. Dyn. Syst., 11 (2004), 393-411. doi: 10.3934/dcds.2004.11.393.  Google Scholar

show all references

##### References:
 [1] R. Beals, D. Scattinger and J. Szmigielski, Acoustic scatting and the extended Korteweg-de Vries hierarchy, Adv. Math., 140 (1998), 190-206. doi: 10.1006/aima.1998.1768.  Google Scholar [2] 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 [3] A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl. (Singap.), 5 (2007), 1-27.  Google Scholar [4] 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 [5] R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. doi: 10.1016/S0065-2156(08)70254-0.  Google Scholar [6] J.-Y. Chemin, "Perfect Incompressible Fluids," in "Oxford Lecture Series in Mathematics and its Applications", Vol. 14, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar [7] G. M. Coclite, H. Holden and K. H. Karlsen., Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM J. Math. Anal., 37 (2006), 1044-1069. doi: 10.1137/040616711.  Google Scholar [8] A. Constantin, The Hamiltonian structure of the Camassa-Holm equation, Exposition. Math., 15 (1997), 53-85.  Google Scholar [9] A. Constantin, Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362. doi: 10.5802/aif.1757.  Google Scholar [10] A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. Roy. Soc. London A, 457 (2001), 953-970. doi: 10.1098/rspa.2000.0701.  Google Scholar [11] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.  Google Scholar [12] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 1992 (2009), 165-186. doi: 10.1007/s00205-008-0128-2.  Google Scholar [13] A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Annali Sc. Norm. Sup. Pisa, 26 (1998), 303-328.  Google Scholar [14] 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 [15] 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 [16] A. Constantin and J. Escher, On the blow-up rate and the blow-up of breaking waves for a shallow water equation, Math. Z., 233 (2000), 75-91. doi: 10.1007/PL00004793.  Google Scholar [17] A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431. doi: 10.1090/S0273-0979-07-01159-7.  Google Scholar [18] A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12.  Google Scholar [19] 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 [20] 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>3.3.CO;2-C.  Google Scholar [21] H. Dai, Model equations for nolinear dispersive waves in compressible Mooney-Rivlin rod, Acta Mech., 127 (1998), 193-207. doi: 10.1007/BF01170373.  Google Scholar [22] R. Dachin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988.  Google Scholar [23] R. Dachin, A note on well-posedness for Camassa-Holm equation, J. Differential Equations, 192 (2003), 429-444. doi: 10.1016/S0022-0396(03)00096-2.  Google Scholar [24] R. Dachin, "Fourier Analysis Methods for PDEs," Lecture Notes, 14 November 2003. Google Scholar [25] H. R. Dullin, G. A. Gottwald and D. D. Holm, An integral shallow water equation with linear and nonlinear dispersion, Phys. Rev. Lett., 87 (2001), 4501-4504. doi: 10.1103/PhysRevLett.87.194501.  Google Scholar [26] J. Escher and Z. Yin, Initial boundary value problems of the Camassa-Holm equation, Comm. Partial Differential Equations, 33 (2008), 377-395. doi: 10.1080/03605300701318872.  Google Scholar [27] J. Escher and Z. Yin, Initial boundary value problems for nonlinear dispersive wave equations, J. Funct. Anal., 256 (2009), 479-508. doi: 10.1016/j.jfa.2008.07.010.  Google Scholar [28] A. Fokas and B. Fuchssteiner, Symplectic structures, their B$\ddota$cklund transformations and hereditary symmetries, Phys. D, 4 (1981), 47-66.  Google Scholar [29] C. Guan, K. H. Karlsen and Z. Yin, Well-posedness and blow-up phenomena for a modified two-component Camassa-Holm equation, Contemp. Math., 526 (2010), 199-220. doi: 10.1090/conm/526/10382.  Google Scholar [30] C. Guan and Z. Yin, Global weak solutions for a modified two-component Camassa-Holm equation, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 28 (2011), 623-641. doi: 10.1016/j.anihpc.2011.04.003.  Google Scholar [31] G. 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 [32] 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 [33] D. Holm, J. Marsden and T. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81.  Google Scholar [34] D. Holm, L. Naraigh and C. Tronci, Singular solution of a modified two-component Camassa-Holm equation, Phys. Rev. E(3),79 (2009), 1-13.  Google Scholar [35] D. Ionescu-Krus, Variational derivation of the Camassa-Holm shallow water equation, J. Nonlinear Math. Phys., 14 (2007), 303-312. doi: 10.2991/jnmp.2007.14.3.1.  Google Scholar [36] R. Ivanov, Water waves and integrability, Philos. Trans. R. Soc. Lond. Ser. A., Math. Phys. Eng. Sci., 365 (2007), 2267-2280. doi: 10.1098/rsta.2007.2007.  Google Scholar [37] R. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 457 (2002), 63-82.  Google Scholar [38] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in "Spectral Theory and Differential Equations", Lecture Notes in Math., 448, Springer Verlag, Berlin, (1975), 25-70.  Google Scholar [39] 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 [40] X. Liu and Z. Yin, On the low regularity solutions for a modified two-component Camassa-Holm shallow water system, Glasgow. Math. J., 53 (2011), 611-621. doi: 10.1017/S0017089511000176.  Google Scholar [41] J. Marsden and T. Ratiu, "Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanics and Systems," 2nd ed., Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1999.  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] W. Tan and Z. Yin, Global dissipative solutions of a modified two-component Camassa-Holm equation shallow water system, J. Math. Phys., 52 (2011), 1-24. 033507. doi: 10.1063/1.3562928.  Google Scholar [44] W. Tan and Z. Yin, Global periodic conservative solutions of a periodic modified two-component Camassa-Holm equation, J. Funct. Anal., 261 (2011), 1204-1226. doi: 10.1016/j.jfa.2011.04.015.  Google Scholar [45] J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.  Google Scholar [46] G. B. Whitham, "Linear and Nonlinear Waves," J. Wiley and Sons, New York, 1980.  Google Scholar [47] 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>3.3.CO;2-X.  Google Scholar [48] K. Yan and Z. Yin, Analytic solutions of the Cauchy problem for two-component shallow water systems, Math. Z., 269 (2011), 1113-1127. doi: 10.1007/s00209-010-0775-5.  Google Scholar [49] Z. Yin, Well-posedness, blowup, and global existence for an integrable shallow water equation, Discrete Contin. Dyn. Syst., 11 (2004), 393-411. doi: 10.3934/dcds.2004.11.393.  Google Scholar
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