Existence And Asymptotic Behavior Of Solutions For Incompressible Navier-Stokes Equations And Related Coupling Problems

In present years,the research on Navier-Stokes equations and its related coupled problems attracts much attention.In this paper,we handle with the existence and asymptotic behavior of solutions to fluid equations and fluid-structure interaction problem.Specifically,the main content of this thesis is as follows:In Chapter 1,we recall the background of the stochastic Navier-Stokes equations and fluid-structure interaction problems.We analyze and conclude the main work and then give some notations and useful lemmas.In Chapter 2,we consider a stochastic model which describes the motion of a 2D incompress-ible fluid in an unbounded domain with viscosity and memory effects.We first investigate the well-posedness by using the classical Faedo-Galerkin method.Unlike the general method of en-ergy estimate,we then split the solution into two parts and get the low-order and high-order uniform estimates,respectively.Based on the uniform estimates of far-field values of solutions,we further prove the existence and uniqueness of random attractors in unbounded domains with a constructed compact subspace corresponding to memory.Finally,we give the upper semicontinuity of the at-tractors when stochastic perturbation approaches to zero.In Chapter 3,we are concerned with a fluid-structure interaction problem with navier slip boundary conditions in which the fluid is considered as a Non-Newtonian fluid and the structure is established by a nonlinear multi-layered model.The fluid domain is driven by a nonlinear elastic shell and thus is not fixed.Therefore,to analysis the problem,we map the fluid problem into a fixed domain by applying the arbitrary Lagrange Euler mapping.We combine the time-discretization and split the problem into a fluid subproblem and a structure subproblem.Since the structure subprob-lem is nonlinear,Lax-Milgram lemma does not work.So we prove the existence and uniqueness by means of the traditional semigroup theory.Notice that the Non-Newtonian fluid possesses a p-Laplacian structure,we show the existence and uniqueness of fluid subproblem by considering the Browder-Minty theorem.With the uniform energy estimates,we deduce the weak and weak*convergence respectively.By a generalized Aubin-Lions-Simon Lemma,we give the strong con-vergence and pass to the limit.