Asymptotic Behaviour Of Solutions For Incompressible Navier-Stokes Equations And Related Nonlocal Coupling Problems

The three-dimensional incompressible Navier-Stokes equations derived from the linear relationship between the stress and the rate-of-strain in the fluid,are very significant both in a purely mathematical sense and in the fluid applications including physics and biology.This paper considers the long time behaviour of Navier-Stokes equations with infinite delay and general hereditary memory and globally modified nonlocal Cahn-Hilliard-Navier-Stokes systems.The structure of this paper is as follows:In Chapter 1,we mainly introduce the background and development of Navi er-Stokes equations and Cahn-Hilliard-Navier-Stokes systems,and briefly summarize the main work of this paper.Some relevant symbols and common results are presentedIn Chapter 2,we address the dynamics of two-dimensional Navier-Stokes models with infinite delay and hereditary memory,whose kernels are a much larger class of functions than the one considered in the literature,in a bounded domain.We prove the existence and uniqueness of weak solutions by means of classical Faedo-Galerkin method.Moreover,we establish the existence of global attractors for the system with the existence of abounded absorbing set and asymptotic compact property through constructing Lyapunov functional and splitting the systemsIn Chapter 3,we consider the systems of partial differential equations modeling the motion of an incompressible isothermal mixture of two immiscible fluids which consists of globally modified Navier-Stokes equations coupled with a convective nonlocal Cahn-Hilliard equations.While the results about the uniqueness of weak solutions for three dimensional system is an open problem due to the estimates of convective term,we establish the uniqueness of weak solutions for globally modified systems in several situations.Then we address the existence of pullback attractors for the systems in the case of constant viscous viscosity,regular pontential and constant mobilityIn Chapter 4,we summarize the context of this paper and give the prospect of future research...