摘要
本博士学位论文主要分两部分。前一部分(第一章至第六章)研究非牛顿流方程解的渐近行为。后一部分(第七章至第十一章)研究格点系统的解的渐近行为。流体力学现象普遍存在于物理学、生物学、大气与海洋科学及航空工业等领域。非牛顿流体力学是近代流体力学的一个重要分支。论文前一部分主要研究数学家Ladyzhenskaya提出的一个非牛顿流模型,主要证明该非牛顿流轨道吸引子及一致吸引子、后拉吸引子的存在性和正则性。同时证明了具有快速振动外力项的非牛顿流一致吸引子的稳定性及具有时滞项的非牛顿流后拉吸引子的存在性与正则性。离散与连续是客观世界物质运动对立统一的两种形式。格点系统是某些变量离散化的时空系统,包括耦合的常微分方程组、耦合映射格点和细胞自动机。在某些情况下,格点系统表现为偏微分方程的空间变量离散化近似。在论文后一部分中,我们先证明时滞格点系统整体吸引子、核截面与一致吸引子存在的充分必要条件,并将结果应用到时滞格点反应扩散方程,证明了整体吸引子的存在性,上半连续性和极限行为。接着考虑了两个典型数学物理方程(Klein-Gordon-Schr(?)dinger方程和长波-短波共振方程)在无穷格点上紧致核截面的存在性、上半连续性和Kolmogorovε-熵的上界估计。然后考虑了非经典抛物方程和复Ginzburg-Landau方程在无穷格点上的整体吸引子的极限行为。之后,我们证明了Hilbert空间中紧集具有有限分形维数的准则,并将结果应用到具体的格点系统中。最后,我们考虑随机格点系统,证明了随机格点动力系统存在随机整体吸引子的充分条件,并将该条件应用到随机格点sine-Gordon方程上。
论文具体安排如下:
第一章首先概述无穷维动力系统理论的背景,介绍无穷维动力系统相关的概念和主要结果。然后概述非牛顿流的现实背景及当前国际上的研究情况,并概述本文在这一方面所做的工作。最后,我们介绍无穷格点系统的起源与当前国际上的研究概况,并概述本文在这方面所做的主要研究工作。
第二章考虑自治情形非牛顿流的轨道吸引子的存在性。在该方程的解的唯一性没有得到证明的情况下,我们借助作用在轨道空间中的自然平移半群,证明紧致吸收集的存在性,从而证明轨道吸引子的存在性,同时得到了广义整体吸引子的存在性。
第三章考虑非自治非牛顿流的一致吸引子的存在性。我们先通过一些细致的先验估计证明H空间中一致吸引子的存在性,然后应用谱分析的技巧证明V空间中一致吸引子的存在性。最后我们应用一致Gronwall不等式和方程自身的特点证明H空间中一致吸引子与V空间中一致吸引子是相等的,从而得到了一致吸引子的正则性。该正则性揭示了该非牛顿流方程的解的渐近光滑效应:解(具有H~2正则性)会最终变得比初值(具有L~2正则性)更光滑。
第四章研究非自治非牛顿流解的后拉渐近行为。首先,我们通过细致的先验估计证明H空间中后拉吸引子的存在性。然后应用椭圆算子的谱分析的技巧来证明V空间中后拉吸引子的存在性。与第三章相似,我们应用一致Gronwall不等式证明H空间中的后拉吸引子与V空间中的后拉吸引子实际上是相等的。该正则性揭示了该非牛顿流方程解的后拉渐近光滑效应:解在相关环的后拉作用下会变得比初值更光滑。
第五章考虑具有快速振动(关于时间)外力项的的非牛顿流。在适当的假设下,我们证明振动方程与平均方程的一致吸引子之间在Hausdorff距离意义下的逼近关系。
第六章考虑具有时滞的非牛顿流方程。我们证明不同空间上的过程后拉吸引子的存在性。然后应用能量方法证明了这两组后拉吸引子之间的关系,并通过得到关系证明后拉吸引子的正则性。
在第七章,我们先证明时滞格点系统存在整体吸引子的充分必要条件。然后应用该结果证明时滞反应扩散方程在无穷格点上整体吸引子的存在性。接着我们考虑时滞区间长度趋近于零时整体吸引子的奇异极限行为。最后,我们说明对于紧致核截面和一致吸引子有相似的结果成立。
第八章证明Klein-Gordon-Schr(?)dinger方程和长波-短波共振方程在无穷格点上紧致核截面的存在性、上半连续性以及Kolmogorovε-熵的估计。
第九章考虑非经典抛物方程和复Ginzburg-Landau方程在无穷格点上的整体吸引子的奇异(关于方程中的参数)极限行为。我们通过证明解对系统中参数的连续依赖性证明了整体吸引子关于参数的连续依赖性。
在第十章,我们先证明Hilbert空间中紧集具有有限分形维数的一个准则,然后把该准则应用到非自治一阶无穷格点系统得到了有限维核截面的存在性。
第十一章考虑随机格点系统。我们先证明了随机格点动力系统存在随机整体吸引子的充分条件。然后把得到的结果应用到随机格点sine-Gordon方程上得到随机整体吸引子的存在性,并证明该随机整体吸引子的Kolmogorovε-熵的估计。
This dissertation is divided into two parts.The first one(from Chapter 1 to Chapter 6) is related to the asymptotie behavior of solutions for non-Newtonian fluid.The second one(from Chapter 7 to Chapter 11) is concerned with the asymptotic behavior of solutions for lattice systems.Fluid dynamics occurs in physics,biology,atmosphere and ocean.ete. Non-Newtonian fluid is an important branch of modern fluid dynamics.The first part of this dissertation studies the existence and regularity of trajectory attractor,uniform attractor and pullback attractor for the Ladyzhenskaya model on non-Newtonian fluid. Also we prove the stability of the uniform attractor for the non-Newtonian fluid with rapidly oscillating external forces,and establish the existence and regularity of pullback attractor for the non-Newtonian fluid with delays.Discretization and continuity are the two forms of motion for the objective substance.Lattice dynamical systems(LDSs) are the spatiotemporal systems with discretization in some variables including coupled ODEs and coupled map lattices and cellular automata[26,36].In some cases,LDSs occur as spatial discretizations of partial differential equations(PDEs).In the second part of the dissertation,we first establish the sufficient and necessary conditions for the existence of global attractor,kernel sections and uniform attractor for the lattice systems with delays. Then we apply the results to retarded lattice reaction-diffusion equations and verily the existence,upper semicontinuity and limiting behavior of the global attractor.Thirdly,we prove the existence,upper semicontinuity and boundedness of Kohnogorovε-entropy of compact kernel sections for the process associated to Klein-Gordon-Schrodinger equations and long-wave-short-wave resonance equations on infinite lattices.At the same time,we consider the limiting behavior of global attractors for the nonclassical parabolic equation and complex Ginzburg-Landau equation on infinite lattices.We also prove a criteria of finite fractal dimensionality for compact set in Hilbert space and apply this criteria to concrete lattice systems.Finally,we establish a sufficient condition for the existence of stochastic global attractor for stochastic lattice dynamical system and apply this result to the stochastic lattice sine-Gordon equation.
The dissertation is arranged as follows.
In Chapter 1,we first summarize the background of infinite dimensional dynamical system,as well as the concept and classical result related the infinite dimensional dynamical system.Then we introduce the physical significance of the non-Newtonian fluid and summarize the related researching surveys.Also,we summarize the main result on the non-Newtonian fluid within this dissertation.Lastly,we introduce the origin of the infinite lattice systems and the researching surveys,and also we summarize the main result on infinite lattice systems within this dissertation.
In Chapter 2,we consider the existence of trajectory attractor for the non-Newtonian fluid.When the uniqueness of solutions is unknown,we consider the natural translation semigroup acting on the trajectory space and prove the existence of compact absorbing set.Then we establish the existence of trajectory attractor and general global attractor.
In Chapter 3,we consider the existence of uniform attractor for the non-Newtonian fluid.Firstly,we prove the existence of the uniform attractor in space H,via some delicate a priori estimations.Then we use the analysis of spectrum to elliptic operator and establish the existence of the uniform attractor in space V.Lastly,we use the uniform Gronwall inequality and the character of the equations to verify that the obtained two uniform attractors coincide with each other,i.e.,we prove the regularity of the uniform attractor, which implies the asymptotic smoothing effect of the fluid in the sense that the solutions become eventually more regular than the initial data.
In Chapter 4,we consider the existence of pullback attractor for the non-Newtonian fluid.We first prove the existence of the uniform attractor in space H,via some delicate a priori estimations.Then we use the analysis of spectrum to elliptic operator and establish the existence of the pullback attractor in space V.Finally,we use the uniform Gronwall inequality to verify that the obtained two pullback attractors coincide with each other, which reveals the pullback asymptotic smoothing effect of the fluid in the sense that the solutions become eventually more regular than the initial data.
In Chapter 5,we consider the non-Newtonian fluid driven by external forces that are rapidly oscillating in time but have a smooth average.We mainly prove that the uniform attractor of the oscillating equations could be approximated by the one of the averaged equations.
Chapter 6 discusses the non-Newtonian fluid with delays.We first prove the existence of pullback attractor for the process defined on different state space.Then we verify the relation between these pullback attractors via energy method.Finally,we prove the regularity of the pullback attractor by using the obtained relation.
In Chapter 7.we first give a sufficient and necessary condition for the existence or a global attractor associated to retarded lattice systems.Then we apply this result to lattice reaction-diffusion equations and obtain the existence of global attractors,and then we consider the singular limiting behavior of the global attractor as the length of the delayed interval tend to zero.Lastly,we remark that some similar results holds for the compact kernel sections and uniform attractors.
In Chapter 8,we prove the existence,upper semi-continuity and an upper bound of Kolmogorov-εentropy of compact kernel sections for the process associated to Klein-Gordon-Schrodinger equations and long-wave-short-wave resonance equations on infinite lattice.
In Chapter 9.we consider the singular limiting hehavior of the global attractor for the nonclassical parabolic equation and complex Ginzburg-Landau equation on infinite lattice.
In Chapter 10,we prove a criteria of finite fractal dimensionality for compact set in Hilbert space and apply this criteria to first-order lattice systems.
In Chapter 11,we establish a sufficient condition for the existence of stochastic global attractor for stochastic lattice dynamical system and apply this result to the stochastic lattice sine-Gordon equation.
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