两类偏微分方程的不变流形

发布时间：2018-07-11 10:12

本文选题：Hamilton系统 + KAM理论　；参考：《山东大学》2017年博士论文

【摘要】：本文主要研究偏微分方程的不变流形,即中心流形,稳定流形和不稳定流形,其中为定义在Banach空间X内的非线性偏微分算子,(?)为线性算子,其谱具有三分性,N为非线性算子且其算子阶小于(?)的算子阶.在动力系统理论中人们特别关心方程(0.1)的一些特殊解(平衡解,周期解,拟周期解等)的存在性和稳定性,而这些特解周围的其它类型解的性质可以通过研究这些特解而得到.KAM(Kolmogorov-Arnold-Moser)理论是研究偏微分方程拟周期解的一个强有力工具,它主要包括正规形方法和Newton-Nash-Moser隐函数定理方法.用正规形方法得到的拟周期解是线性稳定的.最近Xavier Carbe,Ernest Fontich和Rafael de la Llave引入参数化方法来求解线性算子的谱具有三分性的方程的拟周期解.该方法利用线性算子(?)的(离散)谱的三分性把原系统约化到一个有限维的子空间中.虽然该方法也需要无穷次KAM迭代,但是所解的同调方程均是有限维.而需要指出的是该方法需要借助数值模拟等方法预先求出所考察方程的一个近似解K(t),即其中||e||Y充分小.本文主要考察Boussinesq方程关于以上两个方程已经有大量的研究成果.特别地,Rafael de la Llave在2009证明了(0.2)拥有光滑的中心流形.2015年徐君祥证明了(0.2)在适当的条件下存在一族具有2个频率的拟周期解.2016年Rafael de la Llave和Yannick Sire证明了(0.2)有有限维的拟周期解.2008年K.W.Chug和袁小平证明在d = 1情况下(0.3)存在周期解和2-维的拟周期解.2009年,丛洪滋,刘建军和袁小平证明d1时方程(0.3)也存在2-维拟周期解.2011年丛洪滋和高美娜证明了带导数的复Ginzburg-Landau方程存在2-维拟周期解.2013年程红玉和司建国证明了具有拟周期驱动的复Ginzburg-Landau方程存在(m + 2)-维的拟周期解,此时驱动的频率满足Diophantine条件.本文的具体安排如下:第一章分为五节.第一节介绍所研究问题的背景,特别是两个具体模型的研究背景及研究现状.第二节给出一些常用定义,不等式,引理,命题等.第三节给出有限维和无穷维Hamilton系统的KAM定理.第四节给出具有驱动的非Hamilton系统的KAM定理,其中驱动频率满足Diophantine条件.第五节我们介绍 Xavier Carbe,Ernest Fontich 和 Rafael de la Llave 在 2003 年引入的参数化方法并简单地介绍用该方法构造拟周期解的过程.第二章我们构造具有拟周期驱动(驱动频率满足Diophantine条件)的复Ginzburg-Landau方程的拟周期解.具体地,我们把所研究方程的解写成所在空间基的线性组合,把该和代入方程后得到一个格点方程,然后我们做一个变换消去格点方程中的不可积项,再通过作用量角变量变换和一些简单的计算我们得到一个可积系统加小扰动的系统.最后利用第一章所证明的定理1.5得到我们的结果.第三章我们构造Boussinesq方程和复Ginzburg-Landau方程的有界解附近的稳定流形.值得说明的是该解可以是用任何方法得到的(向前)有界解.具体地,假设u(t)=K(t)是(0.1)的解,我们需要找到另一个函数ξ(t)使得u(t)= K(t)+ξ(t也是(0.1)的解.把是u(t)=K(t)和u(t)=K(t)+ξ(t)分别代入(0.1)中并经简单的计算得到关于Ο(t)的发展方程(原方程的变分方和复 Ginzburg-Landau 方程程).以拟周期解K(θ+ωt)为例,所求它的稳定流形就是集合W我们称W是函数w的图像,这里函数w是满足的函数.需要用压缩不动点定理来求函数w.第四章是附录,一些引理的详细证明在本章给出。
[Abstract]:In this paper, we mainly study the invariant manifolds of partial differential equations, namely, central manifolds, stable manifolds and unstable manifolds. It is a nonlinear partial differential operator defined in the Banach space X, (?) as a linear operator, its spectrum has three points, N is a nonlinear operator and its operator order is smaller than (?) operator order. The existence and stability of some special solutions (equilibrium solutions, periodic solutions, quasi periodic solutions, etc.) of equations (0.1), and the properties of other types of solutions around these special solutions can be obtained by studying these special solutions and the.KAM (Kolmogorov-Arnold-Moser) theory is a powerful tool for studying the quasi periodic solutions of partial differential equations, which mainly includes normal forms. Methods and the method of Newton-Nash-Moser implicit function theorem. The quasi periodic solution obtained by the normal form is linear and stable. Recently Xavier Carbe, Ernest Fontich and Rafael de la Llave introduce the parameterized method to solve the quasi periodic solution of the linear operator's spectrum with three partial equations. This method uses three of the (Discrete) spectrum of linear operator (?) The original system is reduced to a finite dimensional subspace. Although the method also needs infinite KAM iteration, the homology equation of the solution is finite dimension. It is necessary to point out that the method needs to predict an approximate solution K (T) of the equation by means of numerical simulation. In other words, the ||e||Y is sufficiently small. The Boussinesq equation has a large number of research results on the above two equations. In particular, Rafael de la Llave in 2009 proves that (0.2) having a smooth center manifold,.2015 years Xu Junxiang proved (0.2) there is a family of quasi periodic solutions of.2016 years Rafael de la Llave and Yannick 0.2 under appropriate conditions (0.2) ) there is a finite dimensional quasi periodic solution.2008 years K.W.Chug and Yuan Xiaoping proof that under the d = 1 case (0.3) the existence of periodic solutions and 2- dimensional quasi periodic solutions for.2009 years, bushes, Liu Jianjun and Yuan Xiaoping prove D1 time equation (0.3) also exist the 2- dimensional quasi periodic solution.2011 year cluster, and Gao Meina prove that the complex Ginzburg-Landau equation with derivative exists 2- dimension. The periodic solution.2013 years Cheng Hongyu and the founding of the nation prove that the quasi periodic driven complex Ginzburg-Landau equation has a quasi periodic solution of (M + 2) - dimension, and the frequency of the drive satisfies the Diophantine condition. The specific arrangement in this paper is as follows: the first chapter is divided into five sections. The first section introduces the background of the research problems, especially the study of the two specific models. The second section gives some common definitions, inequalities, lemma, propositions and so on. The third section gives the KAM theorem of finite and infinite dimensional Hamilton systems. The fourth section gives a KAM theorem for a non Hamilton system with a driving force, in which the driving frequency satisfies the Diophantine condition. Fifth we introduce Xavier Carbe, Ernest Fontich The parameterization method introduced by Rafael de la Llave in 2003 and briefly introduces the process of constructing quasi periodic solutions by this method. In the second chapter, we construct a quasi periodic solution of a complex Ginzburg-Landau equation with quasi periodic drive (driving frequency satisfying Diophantine conditions). The linear combination of the base, after which the sum is replaced by a lattice equation, and then we do a transformation to eliminate the non integrable term in the lattice equation, and then we get a system with small perturbations by the transformation of the action angle variable and some simple calculations. Finally, we get our theorem 1.5 by the theorem of the first chapter. Results. In the third chapter, we construct the stable manifold near the bounded solution of the Boussinesq equation and the complex Ginzburg-Landau equation. It is worth explaining that the solution can be a (forward) bounded solution obtained by any method. In particular, assuming that u (T) =K (T) is (0.1) solution, we need to find another function, (T), to make u (T) = K (T) + zeta (t is (0.1)). U (T) =K (T) and U (T) =K (T) + zeta (T) are substituted in (0.1) respectively, and the equation of development (the variational square and the complex Ginzburg-Landau equation of the original equation) is obtained by simple calculation. The quasi periodic solution K (theta + Omega) is an example, and the stable manifold is the set of images which we call the function. Using compression fixed point theorem to find functions W. the fourth chapter is appendix, and the detailed proof of lemmas is given in this chapter.
【学位授予单位】：山东大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175.2

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