几个流体动力学方程的渐近行为

发布时间：2018-04-06 02:37

  本文选题：无穷维动力系统　切入点：全局吸引子　出处：《安徽大学》2016年博士论文

【摘要】：无穷维动力系统是一门具有广泛应用背景的学科.它主要考虑从物理、化学、流体力学、生命科学以及大气科学等自然科学中大量涌现出来的一些非线性耗散型发展方程解的整体存在性与长时间渐近行为.对于这些耗散系统渐近行为的研究,一方面能帮助我们理解系统的发展演化规律,另一方面还能在一定程度上帮助我们预测系统解的长时间行为,具有重要的理论和实际意义.本篇博士论文主要从无穷维动力系统的角度研究流体动力学中的几个发展方程解的整体存在性和长时间渐近行为.本篇论文共分为六章.在第一章中,我们简单综述无穷维动力系统的基本问题和研究进展.重点阐述自治系统的全局吸引子,指数吸引子理论,非自治系统的一致吸引子、拉回吸引子理论,以及吸引子的分形维数估计理论.第二章中,简单给出一些本文涉及到的函数空间和一些要用到的不等式.在第三章中,考虑三维空间上的三阶梯度流方程解的稳定性问题,证明了解的全局稳定性和渐近稳定性结果,改进了已有文献中的一些结果.在第四章中,考虑有界区域上具有周期边界条件的三阶梯度流方程解的渐近行为.这里直接考虑更为复杂的非自治情形(相对于自治情形).在关于外力项和参数α,β的适当假设下,证明了有界区域上具有周期边界条件的二、三维三阶梯度流体方程具有一致吸引子.进一步,考虑了二维情形下上述三阶梯度流方程的弱解及一致吸引子的稳定性问题.证明了参数α,β趋于零时,上述三阶梯度流方程的弱解和一致吸引子分别收敛到Navier-Stokes方程的弱解和一致吸引子.在第五章中,考察三维有界区域上具有周期边界条件的三阶梯度磁流体方程解的整体存在性和渐近行为首先利用Galerkin逼近和适当的能量估计给出弱解的整体存在性及正则解的存在、唯一性.进一步,利用短轨道方法证明了上述系统在适当空间中具有有限维的全局吸引子及指数吸引子.第六章中,我们考虑有界区域上具有周期边界条件的一类广义的Navier-Stokes方程整体解的存在性与渐近行为.其中3/4α≤1,FN(r)=min{1,N/r},(?)r∈R+.在关于初始值和外力项适当的正则性假设下,证明了上述方程整体解的存在、唯一性.进一步,证明了解半群在适当空间中的全局吸引子存在性,并给出了其分形维数上界的估计.
[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 伪, 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 伪 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 伪 鈮，

本文编号：1717621