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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
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##### References:
 [1] T. Tachim Medjo. A Cahn-Hilliard-Navier-Stokes model with delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2663-2685. doi: 10.3934/dcdsb.2016067 [2] Laurence Cherfils, Madalina Petcu. On the viscous Cahn-Hilliard-Navier-Stokes equations with dynamic boundary conditions. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1419-1449. doi: 10.3934/cpaa.2016.15.1419 [3] T. Tachim Medjo. Robust control of a Cahn-Hilliard-Navier-Stokes model. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2075-2101. doi: 10.3934/cpaa.2016028 [4] T. Tachim Medjo. The exponential behavior of a stochastic Cahn-Hilliard-Navier-Stokes model with multiplicative noise. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1117-1138. doi: 10.3934/cpaa.2019054 [5] Vladimir V. Chepyzhov, E. S. Titi, Mark I. Vishik. On the convergence of solutions of the Leray-$\alpha$ model to the trajectory attractor of the 3D Navier-Stokes system. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 481-500. doi: 10.3934/dcds.2007.17.481 [6] Zhilei Liang. Convergence rate of solutions to the contact discontinuity for the compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1907-1926. doi: 10.3934/cpaa.2013.12.1907 [7] Shuguang Shao, Shu Wang, Wen-Qing Xu, Bin Han. Global existence for the 2D Navier-Stokes flow in the exterior of a moving or rotating obstacle. Kinetic & Related Models, 2016, 9 (4) : 767-776. doi: 10.3934/krm.2016015 [8] Peter Anthony, Sergey Zelik. Infinite-energy solutions for the Navier-Stokes equations in a strip revisited. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1361-1393. doi: 10.3934/cpaa.2014.13.1361 [9] Ciprian Foias, Ricardo Rosa, Roger Temam. Topological properties of the weak global attractor of the three-dimensional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1611-1631. doi: 10.3934/dcds.2010.27.1611 [10] Yong Yang, Bingsheng Zhang. On the Kolmogorov entropy of the weak global attractor of 3D Navier-Stokes equations:Ⅰ. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2339-2350. doi: 10.3934/dcdsb.2017101 [11] Alain Miranville, Giulio Schimperna. On a doubly nonlinear Cahn-Hilliard-Gurtin system. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 675-697. doi: 10.3934/dcdsb.2010.14.675 [12] Hammadi Abidi, Taoufik Hmidi, Sahbi Keraani. On the global regularity of axisymmetric Navier-Stokes-Boussinesq system. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 737-756. doi: 10.3934/dcds.2011.29.737 [13] Irena Pawłow, Wojciech M. Zajączkowski. Regular weak solutions to 3-D Cahn-Hilliard system in elastic solids. Conference Publications, 2007, 2007 (Special) : 824-833. doi: 10.3934/proc.2007.2007.824 [14] Yinghua Li, Shijin Ding, Mingxia Huang. Blow-up criterion for an incompressible Navier-Stokes/Allen-Cahn system with different densities. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1507-1523. doi: 10.3934/dcdsb.2016009 [15] Shixing Li, Dongming Yan. On the steady state bifurcation of the Cahn-Hilliard/Allen-Cahn system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3077-3088. doi: 10.3934/dcdsb.2018301 [16] Alain Miranville. Existence of solutions for Cahn-Hilliard type equations. Conference Publications, 2003, 2003 (Special) : 630-637. doi: 10.3934/proc.2003.2003.630 [17] Álvaro Hernández, Michał Kowalczyk. Rotationally symmetric solutions to the Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 801-827. doi: 10.3934/dcds.2017033 [18] Joanna Rencławowicz, Wojciech M. Zajączkowski. Global regular solutions to the Navier-Stokes equations with large flux. Conference Publications, 2011, 2011 (Special) : 1234-1243. doi: 10.3934/proc.2011.2011.1234 [19] Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085 [20] Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations II: Global existence of small solutions. Evolution Equations & Control Theory, 2012, 1 (1) : 217-234. doi: 10.3934/eect.2012.1.217

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