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September  2019, 18(5): 2283-2298. doi: 10.3934/cpaa.2019103

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

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, China

Received  September 2017 Revised  September 2017 Published  April 2019

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).

Citation: Bo You. Global attractor of the Cahn-Hilliard-Navier-Stokes system with moving contact lines. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2283-2298. doi: 10.3934/cpaa.2019103
References:
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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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F. Boyer, Atheoretical and numerical model for the study of incompressiblemodel flows, Computers and Fluids, 31 (2002), 41-68.   Google Scholar

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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.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

T. Blesgen, A generalization of the Navier-Stokes equation to two-phase flows, J. Phy. D: Appl. Phys., 32 (1999), 1119-1123.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptot. Anal., 20 (1999), 175-212.   Google Scholar

[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.  Google Scholar

[15]

F. Boyer, Atheoretical and numerical model for the study of incompressiblemodel flows, Computers and Fluids, 31 (2002), 41-68.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

E. B. DussanV, The moving contact line: the slip boundary condition, J. Fluid Mech., 77 (1976), 665-684.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

P. C. Hohenberg and B. I. Halperin, Theory of dynamical critical phenomena, Rev. Modern Phys., 49 (1977), 435-479.   Google Scholar

[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.  Google Scholar

[35]

D. Jasnow and J. Vinals, Coarse-grained description of thermo-capillary flow, Phys. Fluids, 8 (1996), 660-669.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

J. Kim, Phase-field models for multi-component fluid flows, Commun. Comput. Phys., 12 (2012), 613-661.  doi: 10.4208/cicp.301110.040811a.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

J. C. Maxwell, On stresses in rarified gases arising from inequalities of temperature, Philos. Trans. R. Soc. London, 170 (1879), 704-712.   Google Scholar

[44]

T. Tachim. Medjo, Pullback attracots for a non-autonomous Cahn-Hilliard-Navier-Stokes system in 2D, Asymptot. Anal., 90 (2014), 21-51.   Google Scholar

[45]

H. K. Moffatt, Viscous and resistive eddies near a sharp corner, J. Fluid Mech., 18 (1964), 1-18.   Google Scholar

[46]

T. Z. Qian, X. Wang and P. Sheng, Molecular scale contact line hydrodynamics of immiscible flows, Phys. Rev. E, 68 (2003), 016306. Google Scholar

[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.  Google Scholar

[48]

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