The Long-time Behavior Of Solutions And Upper Semi-continuity Of Stationary Statistical Properties For Cahn-Hilliard-Brinkman System

In this doctoral dissertation, we are mainly concerned with the long-time behavior of solutions and upper semi-continuity of stationary statistical proper-ties for the Cahn-Hilliard-Brinkman system modeling a diffuse interface model for phase separation of an isothermal incompressible binary fluid in a Brinkman porous medium in a bounded smooth domain Ω(?)R3 with the boundary (?)Ω. Here M stands for the mobility, the quantities ε, v, η and γ are positive constants that represent the diffuse interface thickness, kinematic viscosity of fluid, the fluid permeability and surface tension parameter, respectively, φ is the difference of the fluid (relative) concentrations, p is the fluid pressure, u is the (averaged) fluid velocity,f is the derivative of a double well potential F(s)=1/4(s2 - 1)2 describing the phase separation, and μ is the chemical potential of the binary mixture which is given by the variational derivative of the following free energy functionalIn Chapter 3, we mainly consider the long-time behavior of solutions for the autonomous Cahn-Hilliard-Brinkman system(i.e.,θ= 0). For this system, S. Bosi-a, M. Conti, M. Grasselli ([22]) have obtained the existence of a global attractor in VI(VI={φ∈H1(Ω):∫Ωφdx=I},I∈R) for the semigroup generated by the autonomous Cahn-Hilliard-Brinkman system. In this chapter, we prove the existence of a global attractor in H4(Ω)∩VI and provide the upper boundedness of the fractal dimension of the global attractor. Firstly, by making a priori estimates of solutions, we get the existence of a bounded absorbing set in H4(Ω)∩VI for the semigroup generated by the autonomous Cahn-Hilliard-Brinkman system. Then, we obtain the uniform compactness in Hs(Ω)∩VI (1< s< 4) of the semigroup by using Sobolev compact embedding theorem. Unfortunately, we cannot prove the continuity of the semigroup in Hs(Ω)∩VI (1< s< 4). In order to overcome this d-ifficulty, inspired by the idea of the norm-to-weak continuous semigroup proposed in [152], we obtain the existence of a global attractor in Hs(Ω)∩VI (1< s< 4) of the semigroup. Thanks to the shortage of the regularity of solutions in more reg-ular phase space, the Sobolev compact embedding theorem is invalid to prove the uniform compactness in H4(Ω)∩VI of the semigroup. To overcome this difficulty, we prove the asymptotic compactness in H4(Ω)∩VI of the semigroup by virtue of asymptotic a priori estimates. Combining the theory of the norm-to-weak con-tinuous semigroup, we prove the existence of a global attractor in H4(Ω)∩VI . Furthermore, we provide the upper boundedness of the fractal dimension of the global attractor for Cahn-Hilliard-Brinkman.In Chapter 4, we also consider the long-time behavior of solutions of the non-autonomous Cahn-Hilliard-Brinkman system(i.e.,θ= 1). We obtain the ex-istence of a pullback attractor in H4(Ω)∩VI for the process associated with the non-autonomous Cahn-Hilliard-Brinkman system which is not only important but also useful to prove the upper semi-continuity of stationary statistical properties for the Cahn-Hilliard-Brinkman system in next chapter. For this system, it is easy to obtain the continuity of the process in VI. In addition, we obtain the existence of a pullback absorbing set in H4(Ω)∩VI for the process by making some priori estimates of solutions. Hence, combining Sobolev compact embedding theorem with the theory of the norm-to-weak continuous process proposed in [151], we can prove the existence of a pullback attractor in Hs(Ω)∩VI (1< s< 4) of the pro-cess. However, it is difficult to obtain the uniform boundedness of fluid velocity in (H1(Ω))3 in this case which is different from the case of the autonomous Cahn-Hilliard-Brinkman system. Fortunately, we can prove the uniform boundedness of fluid velocity in Hloc1(R; (H1(Ω))3). Therefore, we first prove the uniform com-pactness of the fluid velocity in (L3(Ω))3 by Aubin-Lions compactness theorem, by which, we establish asymptotic a priori estimates of solutions in H4(Ω)∩VI. Using this asymptotic a priori estimates, we get the asymptotic compactness in H4(Ω) ∩ VI of the process. Finally, combining this result with the theory of the norm-to-weak continuous process, we prove the existence of a pullback attractor in H4(Ω) ∩ VI.In Chapter 5, we are mainly concern with the stability (upper semi-continuity) of stationary statistical properties for dissipative dynamical systems under smal-1 non-autonomous perturbation. G. Lukaszewicz, J.C. Robinson in [108] have considered the existence of invariant measures for the non-autonomous dissipa-tive dynamical system in a complete separable metric space. X.M. Wang in [147] has established an abstract theory about the upper semi-continuity of stationary statistical properties for dissipative dynamical systems under the autonomous perturbation. Inspired by the results in [108] and [147], we establish an ab-stract result about the upper semi-continuous property of invariant measures for dissipative dynamical systems under small non-autonomous perturbation when the processes associated with the dynamical systems under the non-autonomous perturbations satisfy two natural conditions:uniform dissipation and uniform convergence. Furthermore, we prove that the set of invariant measures obtained by the method in [108] of the processes associated with the dynamical systems under the non-autonomous perturbations is convergent in the probability measure space with respect to the weak star-topology. As a corollary, we also obtain the upper semi-continuous property of invariant measures for the autonomous dissipa-tive dynamical systems under small non-autonomous perturbation. Finally, as an application, we apply these abstract results to the two dimensional Navier-Stokes equations and the Cahn-Hilliard-Brinkman system.