GLOBAL ATTRACTOR OF THE CAHN-HILLIARD-NAVIER-STOKES SYSTEM WITH MOVING CONTACT LINES

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).