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Global attractor of the Cahn-Hilliard-Navier-Stokes system with moving contact lines

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  • This paper is concerned with the long-time behavior of solutions for the Cahn-Hilliard-Navier-Stokes system with moving contact lines. Thanks to the strong coupling at the boundary, it is very difficult to obtain the uniqueness of an energy solution for problem (1)-(3) even in two dimension. To overcome this difficulty, inspired by the idea of Sell's radical approach (see [49]) to the global attractor of the three dimensional Navier-Stokes equations, we prove the closedness of the set $ W $ of all global energy solutions for problem (1)-(3) equipped with some metric such that the $ \omega $-limit set of any bounded subset in $ W $ still stay in $ W, $ which is crucial to prove the existence of a global attractor for problem (1)-(3). In addition, we prove the existence of an absorbing set in $ W $ and the uniform compactness of the semigroup $ S_t $ for problem (1)-(3), which entails the existence of a global attractor in $ W $ for problem (1)-(3).

    Mathematics Subject Classification: Primary: 35B40, 35Q35; Secondary: 37L30.


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  • [1] H. Abels, Long-time behavior of solutions of a Navier-Stokes/Cahn-Hilliard system, Proceedings of the Conference Nonlocal and Abstract Parabolic Equations and their Applications, Bedlewo, Banach Center Publications, 86 (2009), 9-19.  doi: 10.4064/bc86-0-1.
    [2] H. Abels, On a diffusive interface model for two-phase flows of viscous, incompressible fluids with matched densities, Arch. Ration. Mech. Anal., 194 (2009), 463-506.  doi: 10.1007/s00205-008-0160-2.
    [3] H. Abels, Strong well-posedness of a diffuse interface model for a viscous, quasi-incompressible two-phase flow, SIAM J. Math. Anal., 44 (2012), 316-340.  doi: 10.1137/110829246.
    [4] H. Abels and E. Feireisl, On a diffuse interface model for a two-phase flow of compressible viscous fluids, Indiana Univ. Math. J., 57 (2008), 659-698.  doi: 10.1512/iumj.2008.57.3391.
    [5] D. M. AndersonG. B. McFadden and A. A. Wheeler, Diffuse-interface methods in fluid mechanics, Annu. Rev. Fluid Mech., 30 (1998), 139-165.  doi: 10.1146/annurev.fluid.30.1.139.
    [6] V. E. BadalassiH. D. Ceniceros and S. Banerjee, Computation of multiphase systems with phase field models, J. Comput. Phys., 190 (2003), 371-397.  doi: 10.1016/S0021-9991(03)00280-8.
    [7] J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci., 7 (1997), 475-502.  doi: 10.1007/s003329900037.
    [8] K. BaoY. ShiS. Sun and X. P. Wang, A finite element method for the numerical solution of the coupled Cahn-Hilliard and Navier-Stokes system for moving contact line problems, J. Comput. Phys., 231 (2012), 8083-8099.  doi: 10.1016/j.jcp.2012.07.027.
    [9] A. BertiV. Berti and D. Grandi, Well-posedness of an isothermal diffusive model for binary mixtures of incompressible fluids, Nonlinearity, 24 (2011), 3143-3164.  doi: 10.1088/0951-7715/24/11/008.
    [10] T. Blesgen, A generalization of the Navier-Stokes equation to two-phase flows, J. Phy. D: Appl. Phys., 32 (1999), 1119-1123. 
    [11] S. Bosia and S. Gatti, Pullback exponential attractor for a Cahn-Hilliard-Navier-Stokes system in 2D, Dyn. Partial Differ. Equ., 11 (2014), 1-38.  doi: 10.4310/DPDE.2014.v11.n1.a1.
    [12] S. BosiaM. Grasselli and A. Miranville, On the longtime behavior of a 2D hydrodynamic model for chemically reacting binary fluid mixtures, Math. Methods Appl. Sci., 37 (2014), 726-743.  doi: 10.1002/mma.2832.
    [13] F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptot. Anal., 20 (1999), 175-212. 
    [14] F. Boyer, Nonhomogenous Cahn-Hilliard fluids, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 225-259.  doi: 10.1016/S0294-1449(00)00063-9.
    [15] F. Boyer, Atheoretical and numerical model for the study of incompressiblemodel flows, Computers and Fluids, 31 (2002), 41-68. 
    [16] C. S. Cao and C. G. Gal, Global solutions for the 2D Navier-Stokes-Cahn-Hilliard model for a two-phase flow of viscous, incompressible fluids with mixed partial viscosity and mobility, Nonlinearity, 25 (2012), 3211-3234.  doi: 10.1088/0951-7715/25/11/3211.
    [17] R. Chella and J. Vinals, Mixing of a two-phase fluid by a cavity flow, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, 53 (1996), 3832-3840. 
    [18] A. Cheskidov and C. Foias, On global attractors of the 3D Navier-Stokes equations, J. Differential Equations, 231 (2006), 714-754.  doi: 10.1016/j.jde.2006.08.021.
    [19] N. J. Cutland and H. J. Keisler, Attractors and neo-attractors for 3D stochastic Navier-Stokes equations, Stoch. Dyn., 5 (2005), 487-533.  doi: 10.1142/S0219493705001559.
    [20] E. B. DussanV, The moving contact line: the slip boundary condition, J. Fluid Mech., 77 (1976), 665-684. 
    [21] E. B. DussanV and S. H. Davis, On the motion of a fluid-fluid interface along a solid surface, J. Fluid Mech., 65 (1974), 71-95. 
    [22] X. B. Feng, Fully discrete element approximations of the Navier-Stokes-Cahn-Hilliard diffuse interface model for two-phase fluid flows, SIAM J. Numer. Anal., 44 (2006), 1049-1072.  doi: 10.1137/050638333.
    [23] X. B. FengY. N. He and C. Liu, Analysis of finite element approximations of a phase field model for two-phase fluids, Math. Comp., 76 (2007), 539-571.  doi: 10.1090/S0025-5718-06-01915-6.
    [24] C. G. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 401-436.  doi: 10.1016/j.anihpc.2009.11.013.
    [25] C. G. Gal and M. Grasselli, Trajectory attractors for binary fluid mixtures in 3D, Chin. Ann. Math. Ser. B, 31 (2010), 655-678.  doi: 10.1007/s11401-010-0603-6.
    [26] C. G. Gal and M. Grasselli, Instability of two-phase flows: a lower bound on the dimension of the global attractor of the Cahn-Hilliard-Navier-Stokes system, Phys. D, 240 (2011), 629-635.  doi: 10.1016/j.physd.2010.11.014.
    [27] C. G. GalM. Grasselli and A. Miranville, Cahn-Hilliard-Navier-Stokes system with moving contact lines, Calc. Var. Partial Differential Equations, 55 (2016), 1-47.  doi: 10.1007/s00526-016-0992-9.
    [28] M. Gao and X. P. Wang, A gradient stable scheme for a phase field model for the moving contact line problem, J. Comput. Phys., 231 (2012), 1372-1386.  doi: 10.1016/j.jcp.2011.10.015.
    [29] M. E. GurtinD. Polignone and J. Vinals, Two-phase binary fluids and immiscible fluids described by an order parameter, Math. Models Methods Appl. Sci., 6 (1996), 815-831.  doi: 10.1142/S0218202596000341.
    [30] M. Heida, On the derivation of thermodynamically consistent boundary conditions for the Cahn-Hilliard-Navier-Stokes system, Internat. J. Engrg. Sci., 62 (2013), 126-156.  doi: 10.1016/j.ijengsci.2012.09.005.
    [31] M. HeidaJ. Málek and K. R. Rajagopal, On the development and generalizations of Cahn-Hilliard equations within a thermodynamic framework, Z. Angew. Math. Phys., 63 (2012), 145-169.  doi: 10.1007/s00033-011-0139-y.
    [32] M. Hintermuller and D. Wegner, Optimal control of a semidiscrete Cahn-Hilliard-Navier-Stokes system, SIAM J. Control Optim., 52 (2014), 747-772.  doi: 10.1137/120865628.
    [33] P. C. Hohenberg and B. I. Halperin, Theory of dynamical critical phenomena, Rev. Modern Phys., 49 (1977), 435-479. 
    [34] D. Jacqmin, Calculation of two-phase Navier-Stokes flows using phase-field modelling, J. Comput. Phys., 155 (1999), 96-127.  doi: 10.1006/jcph.1999.6332.
    [35] D. Jasnow and J. Vinals, Coarse-grained description of thermo-capillary flow, Phys. Fluids, 8 (1996), 660-669. 
    [36] D. KayV. Styles and R. Welford, Finite element approximation of a Cahn-Hilliard-Navier-Stokes system, Interfaces Free Bound, 10 (2008), 15-43.  doi: 10.4171/IFB/178.
    [37] D. Kay and R. Welford, Efficient numerical solution of Cahn-Hilliard-Navier-Stokes fluids in 2D, SIAM J. Sci. Comput., 29 (2007), 2241-2257.  doi: 10.1137/050648110.
    [38] J. Kim, Phase-field models for multi-component fluid flows, Commun. Comput. Phys., 12 (2012), 613-661.  doi: 10.4208/cicp.301110.040811a.
    [39] J. KimK. Kang and J. Lowengrub, Conservative multigrid methods for Cahn-Hilliard fluids, J. Comput. Phys., 193 (2004), 511-543.  doi: 10.1016/j.jcp.2003.07.035.
    [40] A. G. LamorgeseD. Molin and R. Mauri, Phase field approach to multiphase flow modeling, Milan J. Math., 79 (2011), 597-642.  doi: 10.1007/s00032-011-0171-6.
    [41] C. Liu and J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Phys. D, 179 (2003), 211-228.  doi: 10.1016/S0167-2789(03)00030-7.
    [42] J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454 (1998), 2617-2654.  doi: 10.1098/rspa.1998.0273.
    [43] J. C. Maxwell, On stresses in rarified gases arising from inequalities of temperature, Philos. Trans. R. Soc. London, 170 (1879), 704-712. 
    [44] T. Tachim. Medjo, Pullback attracots for a non-autonomous Cahn-Hilliard-Navier-Stokes system in 2D, Asymptot. Anal., 90 (2014), 21-51. 
    [45] H. K. Moffatt, Viscous and resistive eddies near a sharp corner, J. Fluid Mech., 18 (1964), 1-18. 
    [46] T. Z. Qian, X. Wang and P. Sheng, Molecular scale contact line hydrodynamics of immiscible flows, Phys. Rev. E, 68 (2003), 016306.
    [47] T. Z. QianX. Wang and P. Sheng, A variational approach to moving contact line hydrodynamics, J. Fluid Mech., 564 (2006), 333-360.  doi: 10.1017/S0022112006001935.
    [48] R. Ruiz and D. R. Nelson, Turbulence in binary fluid mixtures, Phys. Rev. A: At, Mol. Opt. Phys., 23 (1981), 3224-3246. 
    [49] G. R. Sell, Global attractors for the three-dimensional Navier-Stokes equations, J. Dynam. Differential Equations, 8 (1996), 1-33.  doi: 10.1007/BF02218613.
    [50] J. Shen and X. F. Yang, Energy stable schemes for Cahn-Hilliard phase-field model of two-phase incompressible flows, Chin. Ann. Math. Ser. B, 31 (2010), 743-758.  doi: 10.1007/s11401-010-0599-y.
    [51] J. ShenX. F. Yang and H. J. Yu, Efficient energy stable numerical schemes for a phase field moving contact line model, J. Comput. Phys., 284 (2015), 617-630.  doi: 10.1016/j.jcp.2014.12.046.
    [52] Y. ShiK. Bao and X. P. Wang, 3D adaptive finite element method for a phase field model for the moving contact line problems, Inverse Probl. Imaging, 7 (2013), 947-959.  doi: 10.3934/ipi.2013.7.947.
    [53] E. D. Siggia, Late stages of spinodal decomposition in binary mixtures, Phys. Rev. A: At, Mol. Opt. Phys., 20 (1979), 595-605. 
    [54] V. N. Starovoitov, On the motion of a two-component fluid in the presence of capillary forces, Math. Notes, 62 (1997), 244-254.  doi: 10.1007/BF02355911.
    [55] Z. J. TanK. M. Lim and B. C. Khoo, An adaptive mesh redistribution method for the incompressible mixture flows using phase-field model, J. Comput. Phys., 225 (2007), 1137-1158.  doi: 10.1016/j.jcp.2007.01.019.
    [56] X. P. Wang and Y. G. Wang, The sharp interface limit of a phase field model for moving contact line problem, Methods Appl. Anal., 14 (2007), 287-294.  doi: 10.4310/MAA.2007.v14.n3.a6.
    [57] P. T. YueC. F. Zhou and J. J. Feng, Sharp-interface limit of the Cahn-Hilliard model for moving contact lines, J. Fluid Mech., 645 (2010), 279-294.  doi: 10.1017/S0022112009992679.
    [58] L. Y. ZhaoH. Wu and H. Y. Huang, Convergence to equilibrium for a phase-field model for the mixture of two viscous incompressible fluids, Commun. Math. Sci., 7 (2009), 939-962. 
    [59] Y. Zhou and J. S. Fan, The vanishing viscosity limit for a 2D Cahn-Hilliard-Navier-Stokes system with a slip boundary condition, Nonlinear Anal. Real World Appl., 14 (2013), 1130-1134.  doi: 10.1016/j.nonrwa.2012.09.003.
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