The Dynamical Behavior Of The Nonhomogeneous Nonautonomous 2D Navier-Stokes Equations

The main purpose of this thesis is to consider the existence of uniform (or pullback) attractors of two-dimensional nonautonomous Navier-Stokes equations with nonhomogeneous boundary condition.This thesis consists of six chapters.In the first Chapter we introduce some development of the infinite dimensional dynamical systems in recent years and the main results of this thesis are given.In the second Chapter we present some general tools and basic results of functional analysis which will be used frequently in the thesis.In the third Chapter using the measure of noncompactness, we establish some necessary and sufficient conditions for the existence of the uniform attractors of nonautonomous dynamical systems and give also a new method for proving the existence of the uniform attractors.In the four Chapter we consider the uniform attractors for the two dimensional nonautonomous Navier-Stokes equations in nonsmooth bounded Lip-schitz domain Ω with nonhomogeneous boundary condition u = φ on (?)Ω. Assuming f = f(x,t) ∈ L_loc²((?);D(A^a/4 )), α = -1 or - 2, which is translation compact or normal function and φ ∈ L^∞ ((?)Ω), we establish the existence of the uniform attractor in L²(Ω) and D(A^1/4).In the fifth Chapter we study the existence of compact pullback attractor for the nonautonomous 2D-Navier-Stokes equations in nonsmooth bounded Lipschitz domain with nonhomogeneous boundary condition u = φ. We apply new method to nonhomogeneous nonautonomous Navier-Stokes equation with external forces f(x, t) in L_loc² ((?);E) which is translation bounded. To obtain compact pullback attractor, some abstract results are established. We give a characterization by the concept of measure of noncompactness as well as a method to verify it. We obtain the existence of the pullback attractor in L²(Ω) and D(A^1/4 .In Chapter six using ω-limit compact method, we study the existence of the uniform attractor for the nonautonomous 2D Navier-Stokes equations insome unbounded domain fi with nonhomogeneous boundary condition