Attractors Of Higher-Order Anisotropic Cahn-Hilliard-Navier-Stokes Systems

Cahn-Hilliard-Navier-Stokes systems are important mathematical models de-scribing the evolution of the imcompressible isothermal mixture of binary fluids.This thesis is concerned with attractors of higher-order anisotropic Cahn-Hilliard-Navier-Stokes systems.The main contents of this thesis contain the following three parts.In Part 1,the definition of weak solutions is given and the existence of weak solutions in corresponding space is obtained by Galerkin approximation scheme.Furthermore,the unique-ness of weak solutions to the system is proved by energy estimates.In Part 2,the continuous operator semigroup S(t)is defined and the bounded absorbing sets in Y and V ×H2k are obtained.Then the uniform compactness of operator semigroup is obtained by Sobolev embedding theorem.Therefore,it is proved that higher-order anisotropic Cahn-Hilliard-Navier-Stokes systems possess a global attractor A.Furthermore,A is bounded in H2 ×H2k+2 if region boundary Γ∈ C2k+2.In Part 3,according to the existence theorem of the exponential attractor,the existence of the exponential attractor of higher-order anisotropic Cahn-Hilliard-Navier-Stokes systems is obtained by combining some energy estimates of the uniqueness of weak solutions with the global attractor.