本文選題：無窮維動(dòng)力系統(tǒng)　切入點(diǎn)：全局吸引子　出處：《安徽大學(xué)》2016年博士論文

【摘要】：無窮維動(dòng)力系統(tǒng)是一門具有廣泛應(yīng)用背景的學(xué)科.它主要考慮從物理、化學(xué)、流體力學(xué)、生命科學(xué)以及大氣科學(xué)等自然科學(xué)中大量涌現(xiàn)出來的一些非線性耗散型發(fā)展方程解的整體存在性與長時(shí)間漸近行為.對(duì)于這些耗散系統(tǒng)漸近行為的研究,一方面能幫助我們理解系統(tǒng)的發(fā)展演化規(guī)律,另一方面還能在一定程度上幫助我們預(yù)測(cè)系統(tǒng)解的長時(shí)間行為,具有重要的理論和實(shí)際意義.本篇博士論文主要從無窮維動(dòng)力系統(tǒng)的角度研究流體動(dòng)力學(xué)中的幾個(gè)發(fā)展方程解的整體存在性和長時(shí)間漸近行為.本篇論文共分為六章.在第一章中,我們簡單綜述無窮維動(dòng)力系統(tǒng)的基本問題和研究進(jìn)展.重點(diǎn)闡述自治系統(tǒng)的全局吸引子,指數(shù)吸引子理論,非自治系統(tǒng)的一致吸引子、拉回吸引子理論,以及吸引子的分形維數(shù)估計(jì)理論.第二章中,簡單給出一些本文涉及到的函數(shù)空間和一些要用到的不等式.在第三章中,考慮三維空間上的三階梯度流方程解的穩(wěn)定性問題,證明了解的全局穩(wěn)定性和漸近穩(wěn)定性結(jié)果,改進(jìn)了已有文獻(xiàn)中的一些結(jié)果.在第四章中,考慮有界區(qū)域上具有周期邊界條件的三階梯度流方程解的漸近行為.這里直接考慮更為復(fù)雜的非自治情形(相對(duì)于自治情形).在關(guān)于外力項(xiàng)和參數(shù)α,β的適當(dāng)假設(shè)下,證明了有界區(qū)域上具有周期邊界條件的二、三維三階梯度流體方程具有一致吸引子.進(jìn)一步,考慮了二維情形下上述三階梯度流方程的弱解及一致吸引子的穩(wěn)定性問題.證明了參數(shù)α,β趨于零時(shí),上述三階梯度流方程的弱解和一致吸引子分別收斂到Navier-Stokes方程的弱解和一致吸引子.在第五章中,考察三維有界區(qū)域上具有周期邊界條件的三階梯度磁流體方程解的整體存在性和漸近行為首先利用Galerkin逼近和適當(dāng)?shù)哪芰抗烙?jì)給出弱解的整體存在性及正則解的存在、唯一性.進(jìn)一步,利用短軌道方法證明了上述系統(tǒng)在適當(dāng)空間中具有有限維的全局吸引子及指數(shù)吸引子.第六章中,我們考慮有界區(qū)域上具有周期邊界條件的一類廣義的Navier-Stokes方程整體解的存在性與漸近行為.其中3/4α≤1,FN(r)=min{1,N/r},(?)r∈R+.在關(guān)于初始值和外力項(xiàng)適當(dāng)?shù)恼齽t性假設(shè)下,證明了上述方程整體解的存在、唯一性.進(jìn)一步,證明了解半群在適當(dāng)空間中的全局吸引子存在性,并給出了其分形維數(shù)上界的估計(jì).
[Abstract]:Infinite dimensional dynamic system is a subject with extensive application background.It mainly considers the global existence and long term asymptotic behavior of the solutions of some nonlinear dissipative evolution equations from physics, chemistry, fluid dynamics, life science and atmospheric science.The study of asymptotic behavior of these dissipative systems can help us to understand the evolution law of the system on the one hand, and to a certain extent to predict the long-term behavior of the solution of the system on the other hand, which has important theoretical and practical significance.In this dissertation, the global existence and long term asymptotic behavior of solutions of several evolution equations in hydrodynamics are studied from the point of view of infinite dimensional dynamical systems.This thesis is divided into six chapters.In the first chapter, we briefly review the basic problems and research progress of infinite dimensional dynamical systems.In this paper, the global attractor, exponential attractor theory, uniform attractor theory, pull attractor theory and fractal dimension estimation theory of autonomous systems are discussed.In the second chapter, we give some functional spaces and some inequalities to be used in this paper.In chapter 3, we consider the stability of the solution of the third order gradient flow equation in three dimensional space, prove the global stability and asymptotic stability of the solution, and improve some results in the literature.In chapter 4, the asymptotic behavior of the solutions of the third-order gradient flow equations with periodic boundary conditions is considered.The more complex non-autonomous situation is directly considered here (as opposed to the autonomous case).Under the proper assumptions of the external force term and the parameters 偽, 尾, it is proved that there are uniform attractors for the second and third order gradient fluid equations with periodic boundary conditions in the bounded region.Furthermore, the weak solution and the stability of the uniform attractor of the third-order gradient flow equation are considered in the two-dimensional case.It is proved that the weak solution and uniform attractor of the third-order gradient flow equation converge to the weak solution and uniform attractor of the Navier-Stokes equation, respectively, when the parameters 偽 and 尾 tend to 00:00.In chapter V,In this paper, the global existence and asymptotic behavior of solutions of third-order gradient magnetohydrodynamic equations with periodic boundary conditions in three dimensional bounded region are investigated. Firstly, by using Galerkin approximation and appropriate energy estimation, the global existence of weak solutions and the existence and uniqueness of regular solutions are obtained.Furthermore, it is proved that the global attractor and exponential attractor of the system have finite dimension in the proper space by using the short orbital method.In Chapter 6, we consider the existence and asymptotic behavior of global solutions for a class of generalized Navier-Stokes equations with periodic boundary conditions in bounded regions.Among them, 3 / 4 偽 鈮，

本文編號(hào)：1717621