Cahn-Hilliard-Brinkman系統(tǒng)解的長時間行為及靜態(tài)統(tǒng)計性質上半連續(xù)性問題
發(fā)布時間：2018-09-18 06:37
【摘要】：在這篇博士學位論文中,我們主要考慮如下Brinkman多孔介質中等溫不可壓流體相場分離的耗散界面模型Cahn-Hilliard-Brinkman系統(tǒng)在帶有光滑邊界aΩ的三維有界區(qū)域Ω上解的長時間行為及靜態(tài)統(tǒng)計性質的上半連續(xù)性.其中M為遷移率,E,v,η和γ為正常數(shù),分別表示擴散界面厚度,流體運動粘度,流體滲透率和表面張力參數(shù),φ表示流體(相對)濃度差,p表示流體壓力,u表示(平均)流體流速,g為外力項,f為描述相分離雙井勢F(s)=1/4(s2-1)2的導數(shù),μ為化學勢,它是自由能量泛函的變分導數(shù).在本文第三章中,我們主要考慮自治的Cahn-Hilliard-Brinkman系統(tǒng)(即,θ=0)解的長時間行為.對于這個系統(tǒng),S.Bosia,M.Conti,M.Grasselli在[22]中證明了空問VI(VI={φ∈H1(Ω):(?)Ωφdx=I},I∈R)中全局吸引子的存在性.在這一章中,我們主要考慮該系統(tǒng)在空間H4(Ω)∩V1中全局吸引子的存在性并對其分形維數(shù)進行估計.首先,我們通過對方程的解做正則先驗估計,得到了由自治的Cahn-Hilliard-Brinkman系統(tǒng)產生的解半群在空問H4(Ω)∩VI中有界吸收集的存在性,然后利用Sobolev緊嵌入定理得到了解半群在空間Hs(Ω)∩VI (1s4)中的一致緊性.但是我們得不到解半群在空間Hs(Ω)∩VI (1s4)中的連續(xù)性.為了克服這個困難,我們結合文獻[152]中提出的強弱連續(xù)半群的思想證明了解半群在空間Hs(Ω)∩VI (1s4)中全局吸引子的存在性.由于方程的解沒有更高的正則性先驗估計,因此,我們很難用Sobolev緊嵌入定理來證明解半群在空間H4(Ω)∩VI中的一致緊性.為此,我們首先給出方程的解在空間H4(Ω)∩VI中的一個漸近先驗估計,然后利用這個漸近先驗估計,證明了解半群在空間H4(Ω)∩VI中的漸近緊性.最后,結合強弱連續(xù)半群的思想,證得解半群在空間H4(Ω)∩VI中全局吸引子的存在性.此外,我們還對Cahn-Hilliard-Brinkman系統(tǒng)全局吸引子的分形維數(shù)進行了估計,并得到了其分形維數(shù)的上界.在本文第四章中,我們還考慮了非自治Cahn-Hilliard-Brinkman系統(tǒng)(即,θ=1)解的長時間行為，，得到了由非自治Cahn-Hilliard-Brinkman系統(tǒng)所產生的解過程在空間H4(Ω)∩VI中拉回吸引子的存在性,這不僅有獨立的意義,還為下一章證明帶有非自治攝動的Cahn-Hilliard-Brinkman系統(tǒng)靜態(tài)統(tǒng)計性質的上半連續(xù)性做鋪墊.對于這個系統(tǒng),我們容易得到解過程在空間VI中的連續(xù)性并通過對方程的解做正則先驗估計證得了解過程在空間H4(Ω)∩VI中拉回吸收集的存在性.然后結合Sobolev緊嵌入定理及文獻[151]中提出的強弱連續(xù)過程的思想,得到了解過程在空間Hs(Ω)∩VI (1s4)中拉回吸引子的存在性.然而,為得到H4(Ω)∩VI中拉回吸引子的存在性,不同于自治Cahn-Hilliard-Brinkman系統(tǒng)的情況,我們很難得到流體速度在(H1(Ω))3中的一致有界性,但是我們可以得到流體速度在Hloc1(R;(H1(Ω))3)中的一致有界性.為此,我們利用Aubin-Lions緊定理證明了流體速度在空間(L3(Ω))3中的緊性,并利用這一結果給出方程的解在空間H4(Ω)∩VI中的一個漸近先驗估計,然后利用這個漸近先驗估計證明了解過程在空間H4(Ω)∩VI ¨中的漸近緊性.最后,聯(lián)合強弱連續(xù)過程的思想,證得了解過程在空間H4(Ω)∩VI中拉回吸引子的存在性.在本文第五章中,我們主要考慮帶有非自治小攝動的耗散動力系統(tǒng)靜態(tài)統(tǒng)計性質的穩(wěn)定性,即不變測度的上半連續(xù)性問題.在文獻[108]中,G.Luka-szewicz,J.C.Robinson考慮了完備可分度量空間中非自治耗散動力系統(tǒng)不變測度的存在性X.M.Wang在文獻[147]中考慮了帶有自治小攝動的耗散動力系統(tǒng)靜態(tài)統(tǒng)計性質的上半連續(xù)性.受文獻[108]和[147]的啟發(fā),我們在由非自治攝動的耗散動力系統(tǒng)所產生的解過程族滿足兩個自然的假設條件下：一致耗散性和一致收斂性,證得了帶有非自治小攝動的耗散動力系統(tǒng)不變測度的上半連續(xù)性這一抽象結果.另外,我們還在一定條件下證得了由文獻[108]得到的非自治攝動的耗散動力系統(tǒng)的不變測度集在賦予弱拓撲的概率測度空間中是收斂的,且其極限測度為非攝動的耗散動力系統(tǒng)的不變測度.作為這一抽象結果的推論,我們得到了帶有非自治小攝動的自治耗散動力系統(tǒng)不變測度的上半連續(xù)性.最后,我們將所得抽象結果應用到了二維Navier-Stokes方程組及Cahn-Hilliard-Brinkman系統(tǒng)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.
【學位授予單位】：南京大學
【學位級別】：博士
【學位授予年份】：2016
【分類號】：O175
本文編號：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.
【學位授予單位】：南京大學
【學位級別】：博士
【學位授予年份】：2016
【分類號】：O175
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