Cahn-Hilliard-Brinkman系统解的长时间行为及静态统计性质上半连续性问题
发布时间：2018-09-18 06:37
【摘要】：在这篇博士学位论文中,我们主要考虑如下Brinkman多孔介质中等温不可压流体相场分离的耗散界面模型Cahn-Hilliard-Brinkman系统在带有光滑边界aΩ的三维有界区域Ω上解的长时间行为及静态统计性质的上半连续性.其中M为迁移率,E,v,η和γ为正常数,分别表示扩散界面厚度,流体运动粘度,流体渗透率和表面张力参数,φ表示流体(相对)浓度差,p表示流体压力,u表示(平均)流体流速,g为外力项,f为描述相分离双井势F(s)=1/4(s2-1)2的导数,μ为化学势,它是自由能量泛函的变分导数.在本文第三章中,我们主要考虑自治的Cahn-Hilliard-Brinkman系统(即,θ=0)解的长时间行为.对于这个系统,S.Bosia,M.Conti,M.Grasselli在[22]中证明了空问VI(VI={φ∈H1(Ω):(?)Ωφdx=I},I∈R)中全局吸引子的存在性.在这一章中,我们主要考虑该系统在空间H4(Ω)∩V1中全局吸引子的存在性并对其分形维数进行估计.首先,我们通过对方程的解做正则先验估计,得到了由自治的Cahn-Hilliard-Brinkman系统产生的解半群在空问H4(Ω)∩VI中有界吸收集的存在性,然后利用Sobolev紧嵌入定理得到了解半群在空间Hs(Ω)∩VI (1s4)中的一致紧性.但是我们得不到解半群在空间Hs(Ω)∩VI (1s4)中的连续性.为了克服这个困难,我们结合文献[152]中提出的强弱连续半群的思想证明了解半群在空间Hs(Ω)∩VI (1s4)中全局吸引子的存在性.由于方程的解没有更高的正则性先验估计,因此,我们很难用Sobolev紧嵌入定理来证明解半群在空间H4(Ω)∩VI中的一致紧性.为此,我们首先给出方程的解在空间H4(Ω)∩VI中的一个渐近先验估计,然后利用这个渐近先验估计,证明了解半群在空间H4(Ω)∩VI中的渐近紧性.最后,结合强弱连续半群的思想,证得解半群在空间H4(Ω)∩VI中全局吸引子的存在性.此外,我们还对Cahn-Hilliard-Brinkman系统全局吸引子的分形维数进行了估计,并得到了其分形维数的上界.在本文第四章中,我们还考虑了非自治Cahn-Hilliard-Brinkman系统(即,θ=1)解的长时间行为，，得到了由非自治Cahn-Hilliard-Brinkman系统所产生的解过程在空间H4(Ω)∩VI中拉回吸引子的存在性,这不仅有独立的意义,还为下一章证明带有非自治摄动的Cahn-Hilliard-Brinkman系统静态统计性质的上半连续性做铺垫.对于这个系统,我们容易得到解过程在空间VI中的连续性并通过对方程的解做正则先验估计证得了解过程在空间H4(Ω)∩VI中拉回吸收集的存在性.然后结合Sobolev紧嵌入定理及文献[151]中提出的强弱连续过程的思想,得到了解过程在空间Hs(Ω)∩VI (1s4)中拉回吸引子的存在性.然而,为得到H4(Ω)∩VI中拉回吸引子的存在性,不同于自治Cahn-Hilliard-Brinkman系统的情况,我们很难得到流体速度在(H1(Ω))3中的一致有界性,但是我们可以得到流体速度在Hloc1(R;(H1(Ω))3)中的一致有界性.为此,我们利用Aubin-Lions紧定理证明了流体速度在空间(L3(Ω))3中的紧性,并利用这一结果给出方程的解在空间H4(Ω)∩VI中的一个渐近先验估计,然后利用这个渐近先验估计证明了解过程在空间H4(Ω)∩VI ¨中的渐近紧性.最后,联合强弱连续过程的思想,证得了解过程在空间H4(Ω)∩VI中拉回吸引子的存在性.在本文第五章中,我们主要考虑带有非自治小摄动的耗散动力系统静态统计性质的稳定性,即不变测度的上半连续性问题.在文献[108]中,G.Luka-szewicz,J.C.Robinson考虑了完备可分度量空间中非自治耗散动力系统不变测度的存在性X.M.Wang在文献[147]中考虑了带有自治小摄动的耗散动力系统静态统计性质的上半连续性.受文献[108]和[147]的启发,我们在由非自治摄动的耗散动力系统所产生的解过程族满足两个自然的假设条件下：一致耗散性和一致收敛性,证得了带有非自治小摄动的耗散动力系统不变测度的上半连续性这一抽象结果.另外,我们还在一定条件下证得了由文献[108]得到的非自治摄动的耗散动力系统的不变测度集在赋予弱拓扑的概率测度空间中是收敛的,且其极限测度为非摄动的耗散动力系统的不变测度.作为这一抽象结果的推论,我们得到了带有非自治小摄动的自治耗散动力系统不变测度的上半连续性.最后,我们将所得抽象结果应用到了二维Navier-Stokes方程组及Cahn-Hilliard-Brinkman系统上.
[Abstract]:In this doctoral dissertation, we mainly consider the long-time behavior and semi-continuity of the static statistic properties of the Cahn-Hilliard-Brinkman system in a three-dimensional bounded domain with a smooth boundary a_. M is the mobility, E, V, _. The diffusion interface thickness, fluid viscosity, fluid permeability and surface tension parameters are expressed as positive constants, respectively. The difference of fluid (relative) concentration is represented by phi, the fluid pressure is represented by p, the fluid velocity is expressed by u, the external force term is expressed by g, the derivative of phase separation double well potential F (s) = 1/4 (s2-1) 2, and the chemical potential is expressed by mu. In Chapter 3, we mainly consider the long-time behavior of solutions for autonomous Cahn-Hilliard-Brinkman systems (i.e., theta = 0). For this system, S. Bosia, M. Conti, M. Grasselli prove the existence of global attractors in space VI (VI = {phi < <} H1 (_): (?) _phidx = I}, I < R) in [22]. The existence and fractal dimension of global attractors of the system in space H4(_)V1 are estimated. Firstly, we obtain the existence of bounded absorption set in space H4(_)VI of solution semigroups produced by autonomous Cahn-Hilliard-Brinkman system by making a regular prior estimate of the solution of the equation. Then we obtain the existence of bounded absorption set in space H4(_)VI by using Sobolev compact embedding theorem. The consistency of solution semigroups in space Hs(_)VI(1s4) is obtained. But we can not obtain the continuity of solution semigroups in space Hs(_)VI(1s4). To overcome this difficulty, we prove the existence of global attractors for solution semigroups in space Hs(_)VI(1s4) by combining the idea of strong and weak continuous semigroups proposed in [152]. It is difficult to prove the compactness of the solution Semigroup in space H4(_)VI by using Sobolev compact embedding theorem because there is no higher regular prior estimate for the solution of the equation. For this reason, we first give an asymptotic prior estimate of the solution of the equation in space H4(_)VI, and then prove that the solution semigroup is in space by using this asymptotic prior estimate. The asymptotic compactness in space H4(_)VI. Finally, the existence of global attractors for solution semigroups in space H4(_)VI is proved by combining the idea of strong and weak continuous semigroups. In addition, the fractal dimension of global attractors for Cahn-Hilliard-Brinkman system is estimated and the upper bound of fractal dimension is obtained. We also consider the long-time behavior of the solutions of the nonautonomous Cahn-Hilliard-Brinkman system (i.e., theta=1). We obtain the existence of the pullback attractor for the solution process generated by the nonautonomous Cahn-Hilliard-Brinkman system in space H4(_)VI. This is not only of independent significance, but also a proof for the existence of Cahn-Hilliard-Brinkma system with nonautonomous perturbation in the next chapter. For this system, we can easily obtain the continuity of the solution process in the space VI and prove the existence of the absorption set pulled back by the solution process in the space H4(_)VI by the regular prior estimation of the solution of the equation. In order to obtain the existence of the pullback attractor in H4 (_) VI (1s4), however, unlike the case of autonomous Cahn-Hilliard-Brinkman system, it is difficult to obtain the uniform boundedness of the fluid velocity in (H1 (_)) 3, but we can obtain it. The uniform boundedness of the fluid velocity in Hloc1 (R; (H1 (_)) 3 is obtained. For this reason, we prove the compactness of the fluid velocity in space (L3 (_)) 3 by using Aubin-Lions compact theorem, and give an asymptotic prior estimate of the solution of the equation in space H4 (_) VI, and then prove the solution process in space by using this asymptotic prior estimate. The asymptotic compactness in space H4(_)VI is proved. Finally, the existence of pullback attractors for the solution process in space H4(_)VI is proved by combining the idea of strong and weak continuous processes. In the fifth chapter, we mainly consider the stability of static statistical properties of dissipative dynamical systems with small perturbations, i.e. the semi-continuity of invariant measures. In [108], G. Luka-szewicz, J. C. Robinson considered the existence of invariant measures for non-autonomous dissipative dynamical systems in a complete separable metric space. In [147], G. Luka-szewicz and J. C. Robinson considered the semi-continuity of the static statistical properties of dissipative dynamical systems with autonomous small perturbations. Under two natural assumptions: uniform dissipation and uniform convergence, the semi-continuity of invariant measures for dissipative dynamical systems with small perturbations and nonautonomous perturbations is proved. The set of invariant measures for autonomous perturbed dissipative dynamical systems is convergent in the probability measure space endowed with weak topology, and its limit measure is invariant measures for non-perturbed dissipative dynamical systems. As a corollary of this abstract result, we obtain the upper half continuity of invariant measures for autonomous dissipative dynamical systems with small perturbations. Finally, we apply the abstract results to two-dimensional Navier-Stokes equations and Cahn-Hilliard-Brinkman systems.
【学位授予单位】：南京大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175
本文编号：2247085
[Abstract]:In this doctoral dissertation, we mainly consider the long-time behavior and semi-continuity of the static statistic properties of the Cahn-Hilliard-Brinkman system in a three-dimensional bounded domain with a smooth boundary a_. M is the mobility, E, V, _. The diffusion interface thickness, fluid viscosity, fluid permeability and surface tension parameters are expressed as positive constants, respectively. The difference of fluid (relative) concentration is represented by phi, the fluid pressure is represented by p, the fluid velocity is expressed by u, the external force term is expressed by g, the derivative of phase separation double well potential F (s) = 1/4 (s2-1) 2, and the chemical potential is expressed by mu. In Chapter 3, we mainly consider the long-time behavior of solutions for autonomous Cahn-Hilliard-Brinkman systems (i.e., theta = 0). For this system, S. Bosia, M. Conti, M. Grasselli prove the existence of global attractors in space VI (VI = {phi < <} H1 (_): (?) _phidx = I}, I < R) in [22]. The existence and fractal dimension of global attractors of the system in space H4(_)V1 are estimated. Firstly, we obtain the existence of bounded absorption set in space H4(_)VI of solution semigroups produced by autonomous Cahn-Hilliard-Brinkman system by making a regular prior estimate of the solution of the equation. Then we obtain the existence of bounded absorption set in space H4(_)VI by using Sobolev compact embedding theorem. The consistency of solution semigroups in space Hs(_)VI(1s4) is obtained. But we can not obtain the continuity of solution semigroups in space Hs(_)VI(1s4). To overcome this difficulty, we prove the existence of global attractors for solution semigroups in space Hs(_)VI(1s4) by combining the idea of strong and weak continuous semigroups proposed in [152]. It is difficult to prove the compactness of the solution Semigroup in space H4(_)VI by using Sobolev compact embedding theorem because there is no higher regular prior estimate for the solution of the equation. For this reason, we first give an asymptotic prior estimate of the solution of the equation in space H4(_)VI, and then prove that the solution semigroup is in space by using this asymptotic prior estimate. The asymptotic compactness in space H4(_)VI. Finally, the existence of global attractors for solution semigroups in space H4(_)VI is proved by combining the idea of strong and weak continuous semigroups. In addition, the fractal dimension of global attractors for Cahn-Hilliard-Brinkman system is estimated and the upper bound of fractal dimension is obtained. We also consider the long-time behavior of the solutions of the nonautonomous Cahn-Hilliard-Brinkman system (i.e., theta=1). We obtain the existence of the pullback attractor for the solution process generated by the nonautonomous Cahn-Hilliard-Brinkman system in space H4(_)VI. This is not only of independent significance, but also a proof for the existence of Cahn-Hilliard-Brinkma system with nonautonomous perturbation in the next chapter. For this system, we can easily obtain the continuity of the solution process in the space VI and prove the existence of the absorption set pulled back by the solution process in the space H4(_)VI by the regular prior estimation of the solution of the equation. In order to obtain the existence of the pullback attractor in H4 (_) VI (1s4), however, unlike the case of autonomous Cahn-Hilliard-Brinkman system, it is difficult to obtain the uniform boundedness of the fluid velocity in (H1 (_)) 3, but we can obtain it. The uniform boundedness of the fluid velocity in Hloc1 (R; (H1 (_)) 3 is obtained. For this reason, we prove the compactness of the fluid velocity in space (L3 (_)) 3 by using Aubin-Lions compact theorem, and give an asymptotic prior estimate of the solution of the equation in space H4 (_) VI, and then prove the solution process in space by using this asymptotic prior estimate. The asymptotic compactness in space H4(_)VI is proved. Finally, the existence of pullback attractors for the solution process in space H4(_)VI is proved by combining the idea of strong and weak continuous processes. In the fifth chapter, we mainly consider the stability of static statistical properties of dissipative dynamical systems with small perturbations, i.e. the semi-continuity of invariant measures. In [108], G. Luka-szewicz, J. C. Robinson considered the existence of invariant measures for non-autonomous dissipative dynamical systems in a complete separable metric space. In [147], G. Luka-szewicz and J. C. Robinson considered the semi-continuity of the static statistical properties of dissipative dynamical systems with autonomous small perturbations. Under two natural assumptions: uniform dissipation and uniform convergence, the semi-continuity of invariant measures for dissipative dynamical systems with small perturbations and nonautonomous perturbations is proved. The set of invariant measures for autonomous perturbed dissipative dynamical systems is convergent in the probability measure space endowed with weak topology, and its limit measure is invariant measures for non-perturbed dissipative dynamical systems. As a corollary of this abstract result, we obtain the upper half continuity of invariant measures for autonomous dissipative dynamical systems with small perturbations. Finally, we apply the abstract results to two-dimensional Navier-Stokes equations and Cahn-Hilliard-Brinkman systems.
【学位授予单位】：南京大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175
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