Uniformly Attractors For 3D Non-autonomous Nonlinear Viscous Fluid Equations

In this paper,we consider the long time behavior for the regularization 3D non-autonomous nonlinear viscous fluid equations.Firstly,proved the existence and uniqueness of solutions for the equations,and the uniformly prior estimation of the solutions is given.Secondly,discuss the bounded uniformly absorbing sets of the family of processes{U_f?t,??},f?H?f₀?,which further proves that the process is asymptotically compact and continuous.Finally,on a bounded region,the existence of uniform attractors in H,V,H²is proved and giving the relationship between uniform attractors.On unbounded regions,the existence of uniform attractor in V is proved.The full paper is divided into four parts:In chapter 1,we mainly introduce the background source of the Navier-Stokes equation and the development of infinite dimensional dynamical systems,further proposes the problems studied and the main results of this paper.In chapter 2,the preliminary knowledge needed in this paper,as well as the basic concepts of non-autonomous systems and the existence of uniform attractors are given.Some uniform prior estimates of the solution are given when the external force term belongs to the sign space.In chapter 3,in the bounded region,firstly we give the bounded absorbing sets,and use the Sobolev space tight embedding theorem and Ascoli-Arzela theorem to prove of the family of processes{U_f?t,??},f?H?f₀?is asymptotically tight and continuous in the H,V and H²,thus using semigroup theory to get uniform attractor of the process,finally discussed the relationship between the uniform attractors.In chapter 4,in the unbounded region,this paper uses the energy equation and the functional analysis method to prove that the family of processes {U_f?t,??},f?H?f₀?in V is uniformly asymptotically tight,and further obtained the uniform attactor in V.