THE GLOBAL ATTRACTOR FOR A VISCOUS WEAKLY DISSIPATIVE GENERALIZED TWO-COMPONENT μ-HUNTER-SAXTON SYSTEM

doi:10.1016/S0252-9602(18)30772-0

Abstract

Abstract:

This article is concerned with the existence of global attractor of a weakly dissipative generalized two-component μ-Hunter-Saxton (gμHS2) system with viscous terms. Under the period boundary conditions and with the help of the Galerkin procedure and compactness method, we first investigate the existence of global solution for the viscous weakly dissipative (gμHS2) system. On the basis of some uniformly prior estimates of the solution to the viscous weakly dissipative (gμHS2) system, we show that the semi-group of the solution operator {S(t)}_t ≥ 0 has a bounded absorbing set. Moreover, we prove that the dynamical system {S(t)}_t ≥ 0 possesses a global attractor in the Sobolev space H²(S)×H²(S).

Key words: Generalized two-component μ-Hunter-Saxton system, viscous weakly dissipative, existence, global attractor, period boundary conditions

Lei ZHANG, Bin LIU. THE GLOBAL ATTRACTOR FOR A VISCOUS WEAKLY DISSIPATIVE GENERALIZED TWO-COMPONENT μ-HUNTER-SAXTON SYSTEM[J].Acta mathematica scientia,Series B, 2018, 38(2): 651-672.

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References

[1] Alarcon E A, Iorio R J. The existence of global attractors for a class of nonlinear dissipative evolution equations. Proceedings of the Royal Society of Edinburgh Section A:Mathematics, 2005, 135(5):887-913
[2] Babin A V, Vishik M I. Attractors of Evolution Equations//Attractors of evolution equations. NorthHolland, 1992
[3] de Monvel A B, Shepelsky D. Riemann-Hilbert approach for the Camassa-Holm equation on the line. Comptes Rendus Mathematique, 2006, 343(10):627-632
[4] Beals R, Sattinger D H, Szmigielski J. Multi-peakons and a theorem of Stieltjes. Inverse Problems, 1999, 15(1):L1-L4
[5] Constantin A, McKean H P. A shallow water equation on the circle. Communications on Pure and Applied Mathematics, 1999, 52(8):949-982
[6] Camassa R, Holm D D. An integrable shallow water equation with peaked solitons. Physical Review Letters, 1993, 71(11):1661-1664
[7] Camassa R, Holm D D, Hyman J M. A new integrable shallow water equation. Advances in Applied Mechanics, 1994, 31:1-33
[8] Chen R M, Liu Y. Wave breaking and global existence for a generalized two-component Camassa-Holm system. International Mathematics Research Notices, 2010, 2011(6):1381-1416
[9] Chen M, Zhang Y. A two-component generalization of the Camassa-Holm equation and its solutions. Letters in Mathematical Physics, 2006, 75(1):1-15
[10] Constantin A, Strauss W A. Stability of peakons. Communications on Pure and Applied Mathematics, 2000, 53(5):603-610
[11] Constantin A, Strauss W A. Stability of a class of solitary waves in compressible elastic rods. Physics Letters A, 2000, 270(3):140-148
[12] Constantin A. On the scattering problem for the Camassa-Holm equation//Proceedings of the Royal Society of London A:Mathematical, Physical and Engineering Sciences. The Royal Society, 2001, 457(2008):953-970
[13] Constantin A. On the inverse spectral problem for the Camassa-Holm equation. Journal of Functional Analysis, 1998, 155(2):352-363
[14] Constantin A, Gerdjikov V S, Ivanov R I. Inverse scattering transform for the Camassa-Holm equation. Inverse Problems, 2006, 22(6):2197-2207
[15] Ding D, Tian L. The attractor in dissipative Camassa-Holm equation. Acta Mathematicae Applicatae Sinica, 2004, 27(3):536-545
[16] Dai H H. Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod. Acta Mechanica, 1998, 127(1/4):193-207
[17] Escher J, Lechtenfeld O, Yin Z. Well-posedness and blow-up phenomena for the 2-component CamassaHolm equation. Discrete and continuous dynamical systems, 2007, 19(3):493-513
[18] Falqui G. On a Camassa-Holm type equation with two dependent variables. Journal of Physics A:Mathematical and General, 2005, 39(2):327-342
[19] Fuchssteiner B, Fokas A S. Symplectic structures, their Bäcklund transformations and hereditary symmetries. Physica D:Nonlinear Phenomena, 1981, 4(1):47-66
[20] Wei F, Da-Jun Z. The Hamiltonian structures of μ-equations related to periodic peakons. Chinese Physics Letters, 2013, 30(8):080201
[21] Fuchssteiner B. Some tricks from the symmetry-toolbox for nonlinear equations:generalizations of the Camassa-Holm equation. Physica D:Nonlinear Phenomena, 1996, 95(3/4):229-243
[22] Guan C, Yin Z. Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system. Journal of Differential Equations, 2010, 248(8):2003-2014
[23] Gui G, Liu Y. On the global existence and wave-breaking criteria for the two-component Camassa-Holm system. Journal of Functional Analysis, 2010, 258(12):4251-4278
[24] Gui G, Liu Y. On the Cauchy problem for the two-component Camassa-Holm system. Mathematische Zeitschrift, 2011, 268(1):45-66
[25] Gui G, Liu Y, Zhu M. On the wave-breaking phenomena and global existence for the generalized periodic Camassa-Holm equation. International Mathematics Research Notices, 2011, 2012(21):4858-4903
[26] Guo F, Gao H, Liu Y. On the wave-breaking phenomena for the two-component Dullin-Gottwald-Holm system. Journal of the London Mathematical Society, 2012, 86(3):810-834
[27] Gurevich A V, Zybin K P, Gnedin N Y, et al. Nondissipative gravitational turbulence. Zh Eksp Teor Fiz, 1988, 94:3-25
[28] Hunter J K, Saxton R. Dynamics of director fields. SIAM Journal on Applied Mathematics, 1991, 51(6):1498-1521
[29] Hunter J K, Zheng Y. On a completely integrable nonlinear hyperbolic variational equation. Physica D:Nonlinear Phenomena, 1994, 79(2/4):361-386
[30] Ito M. Symmetries and conservation laws of a coupled nonlinear wave equation. Physics Letters A, 1982, 91(7):335-338
[31] Ivanov R. Two-component integrable systems modelling shallow water waves:the constant vorticity case. Wave Motion, 2009, 46(6):389-396
[32] Johnson R S. Camassa-Holm, Korteweg-de Vries and related models for water waves. Journal of Fluid Mechanics, 2002, 455:63-82
[33] Kolev B. Poisson brackets in Hydrodynamics. Discrete and Continuous Dynamical Systems-Series A, 2007, 19(3):555-574
[34] Khesin B, Lenells J, Misio lek G. Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms. Mathematische Annalen, 2008, 342(3):617-656
[35] Liu J, Yin Z. On the Cauchy problem of a periodic 2-component μ-Hunter-Saxton system. Nonlinear Analysis:Theory, Methods & Applications, 2012, 75(1):131-142
[36] Liu J. The Cauchy problem of a periodic 2-component μ-Hunter-Saxton system in Besov spaces. Journal of Mathematical Analysis and Applications, 2013, 399(2):650-666
[37] Liu J, Yin Z. Global weak solutions for a periodic two-component μ-Hunter-Saxton system. Monatshefte für Mathematik, 2012, 168(3/4):503-521
[38] Lenells J, Misio lek G, Tiǧlay F. Integrable evolution equations on spaces of tensor densities and their peakon solutions. Communications in Mathematical Physics, 2010, 299(1):129-161
[39] Lenells J. Conservation laws of the Camassa-Holm equation. Journal of Physics A:Mathematical and General, 2005, 38(4):869-880
[40] Olver P J, Rosenau P. Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. Physical Review E, 1996, 53(2):1900-1906
[41] Tian L, Fan J. The attractor on viscosity Degasperis-Procesi equation. Nonlinear Analysis:Real World Applications, 2008, 9(4):1461-1473
[42] Tian L, Gao Y. The global attractor of the viscous Fornberg-Whitham equation. Nonlinear Analysis:Theory, Methods & Applications, 2009, 71(11):5176-5186
[43] Tian L, Tian R. The attractor for the two-dimensional weakly damped KdV equation in belt field. Nonlinear Analysis:Real World Applications, 2008, 9(3):912-919
[44] Tian L, Xu Y, Zhou J. Attractor for the viscous two-component Camassa-Holm equation. Nonlinear Analysis:Real World Applications, 2012, 13(3):1115-1129
[45] Tian L, Xu Y. Attractor for a viscous coupled Camassa-Holm equation. Advances in Difference Equations, 2010, 2010(1):512812
[46] Temam R. Infinite-dimensional dynamical systems in mechanics and physics. Springer Science & Business Media, 2012
[47] Wang F, Li F, Chen Q. Wave breaking and global existence for a weakly dissipative generalized twocomponent μ-Hunter-Saxton system. Nonlinear Analysis:Real World Applications, 2015, 23:61-77
[48] Yin Z. On the Structure of Solutions to the Periodic Hunter-Saxton Equation. SIAM Journal on Mathematical Analysis, 2004, 36(1):272-283
[49] Zuo D. A two-component μ-Hunter-Saxton equation. Inverse Problems, 2010, 26(8):085003
[50] Zong X, Sun S. On the global attractor of the two-component π-Camassa-Holm equation with viscous terms. Nonlinear Analysis:Real World Applications, 2014, 20:82-98
[51] Zheng S. Nonlinear evolution equations. CRC Press, 2004