Long Time Behavior Of Solutions For Several Dissipative Incompressible Fluid Models

In this paper,Brinkman-Forchheimer equation,Non-Newtonian micropolar fluid equations and magnetohydrodynamic equations are studied.From the perspective of infinite dimensional dynamical system,we systematically study these nonlinear partial differential equations with energy dissipation,and obtain a series of interesting and novel results,which improve the understanding of these equations.In Chapter 1,we mainly introduce the research background of infinite dimensional dynamical systems,the research status and the main work of this paper.In Chapter 2,we mainly introduce some preparatory knowledge and some symbols used in this paper.From chapter 3 to chapter 6,the main results are described as follows:In Chapter 3,by using the Groinov-Hausdorff distance between two global attractors in different phase spaces,we consider that the long-time dynamic behavior of the solution for the three-dimensional Brinkman Forchheimer equation.Specifically,we obtain the stability of the semi dynamic system induced by the three-dimensional Brinkman Forchheimer equation on the global attractor A and the continuous dependence of the global attractor A under the perturbation of the smooth domain Ω.The novelty of this method is that the phase spaces of the global attractor Aε corresponding to the perturbation problem and the global attractor A corresponding to the original problem can be not disjoint.It does not need to use the method introduced in reference[7]to pull the perturbation problem back to the original domain Ω,and it does not need to use different norms on the extension spaces to compare the attractors of different phase spaces.In Chapter 4,we consider the long time behavior of solutions for two dimensional non-Newtonian micropolar fluid equations in the autonomous case.By using the method of l-trajectories,we overcome the lack of sufficient regularity of the solutions of the nonNewtonian micropolar fluid equations,and prove that the solution semigroup S(t)generated by the equations possesses finite dimensional global attractors and exponential attractors in the phase space H×L2(Ω).In Chapter 5,we consider the pullback asymptotic behavior of solutions for two dimensional non-Newtonian micropolar fluid equations in non-autonomous cases.By using the method of l-trajectories,we overcome the difficulty of verifying the Holder continuity in time of the process {U(t,τ)}t≥τ generated by the non-Newtonian micropolar fluid equations in the original phase space H ×L2(Ω).Firstly,we prove that the process {L(t,τ)}t≥τinduced by the equations possesses a finite dimensional pullback attractor in the trajectory space Xl,then we verify the smoothness of the process {L(t,τ)}t≥τ and construct the pullback exponential attractor in the trajectory space Xl.Furthermore,we obtain that the process {U(t,τ)}t≥τ generated by the non-Newtonian micropolar fluid equations possesses finite dimensional pullback attractors and pullback exponential attractors in the original phase space.In Chapter 6,we study the pullback asymptotic dynamical behavior of 2D nonautonomous MHD equations in a non-smooth domain(Lipschitz bounded domain)with non-homogeneous Dirichlet boundary conditions.Specifically,by using the method introduced by R.M.Brown,P.A.Perry,Z.W.Shen in[21],we transform the non-homogeneous boundary value problem into homogeneous boundary value problem.By using the equivalent norm,we establish a new prior estimate and obtain the bounded pullback D-absorbing set.Then by using the energy method introduced by Ball in[20]and Wang in[135],we obtain the asymptotic compactness of the process {U(t,τ)}t≥τ,and prove the existence of the pullback attractor of the process {U(t,τ)}t≥τ generated by the MHD equations in phase space H.Then we estimate the fractal dimension of the pullback D-attractor in H is bounded.