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

Bo You

doi:10.3934/cpaa.2019103

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2019, Volume 18, Issue 5: 2283-2298. Doi: 10.3934/cpaa.2019103

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dinger-Poisson system involving critical exponent

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

Bo You

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

Received: August 31, 2017

Revised: August 31, 2017

Published: March 2019

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Abstract

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).

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

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