Fourier-Besov空间与振荡积分及其应用

发布时间：2018-01-12 11:03

本文关键词：Fourier-Besov空间与振荡积分及其应用　出处：《浙江大学》2015年博士论文　论文类型：学位论文

【摘要】：调和分析中的方法技巧,系统化的概念或理论,都在方程的实际应用中,发挥巨大的作用.本文主要利用调和分析中的Littlewood-Paley理论综合性的讨论Fourier-Besov空间的定义、性质,并得到其在广义Navier-Stokes方程中的一些应用；此外本文利用振荡积分的理论方法研究了阻尼波动方程的一些估计及适定性.论文分为五个章节,其主要内容安排为：第一章回顾本文课题的研究背景与现状,综合论述Fourier-Besov空间的发展和已有研究成果,广义Navier-Stokes方程的研究情况以及阻尼波动方程的一些进展.并且在比较的基础上,给出本文的主要定理.第二章集中讨论Fourier-Besov空间.简要回顾Littlewood-Paley理论和Besov空间的定义以及一些基本性质.通过与Besov空间类似的方式定义Fourier-Besov空间,从这一定义出发,讨论这个空间的等价形式、与其它空间的关系、包含嵌入、插值等性质.尤其是通过Bony分解的方法,给出Fourier-Besov空间的乘积估计.这一章的内容,也是在后面的两章Fourier-Besov空间的应用中经常要用的.第三章第四章给出Fourier-Besov空间在广义Navier-Stokes方程中的一些应用,综合性的考虑广义Navier-Stokes方程在Fourier-Besov空间中的性质.第三章考虑小初值的全局适定性,并在此基础上证明解关于时间的全局衰减性.尤其是得到了方程在端点情形β=1/2时的一个全局适定性.第四章则研究广义Navier-Stokes方程在Fourier-Besov空间中解的爆破准则以及空间正则性.为证明爆破准则关键在于构造方程在关于时间具有连续性的空间中的解,为此证明了方程在Fourier-Besov空间中另外一种形式的解,而空间正则性则采用Gevrey类的办法.第五章考虑振荡积分在高阶阻尼波动方程中的应用.通过基本解的表达形式,发现其核算子的表现呈现着不同的变化：在低频部分表现为热核算子,而在高频部分是一个振荡积分的形式.因此这一章首先分成三部分估计核的点态估计,从而得到基本解在Lp空间上的估计,并进一步利用这些估计得到方程的一个全局解结果.
[Abstract]:Methods and techniques in harmonic analysis, systematic concepts or theories, are all applied in the practical application of equations. This paper mainly discusses the definition and properties of Fourier-Besov space by using the Littlewood-Paley theory in harmonic analysis. Some applications to the generalized Navier-Stokes equation are obtained. In addition, we use the theory of oscillation integral to study some estimates and suitability of damping wave equation. The thesis is divided into five chapters. The main contents are as follows: the first chapter reviews the research background and current situation of this paper. This paper discusses the development of Fourier-Besov space and the existing research results. The research situation of generalized Navier-Stokes equation and some progress of damped wave equation. And on the basis of comparison. The main theorems of this paper are given. Chapter 2 focuses on Fourier-Besov spaces. A brief review of the definition of Littlewood-Paley theory, Besov space and some. Basic properties. Define Fourier-Besov spaces in a similar way to Besov spaces. From this definition, we discuss the equivalent form of this space, the relationship with other spaces, including the properties of embedding, interpolation, etc., especially through the method of Bony decomposition. The product estimation of Fourier-Besov space is given. The content of this chapter. It is often used in the application of Fourier-Besov space in the following two chapters. Chapter 3, chapter 4th, gives the Fourier-Besov space in the generalized Navier-Stok. Some applications in the es equation. The properties of generalized Navier-Stokes equations in Fourier-Besov space are considered synthetically. In chapter 3, the global fitness of small initial values is considered. On this basis, we prove the global decay of the solution on time. In particular, we obtain a global fitness of the equation in the case of 尾 1 / 2 at the end of the equation. Chapter 4th studies the generalized Navier-Stokes square. In order to prove the blow-up criterion and the regularity of the solution of the equation in Fourier-Besov space, the key is to construct the solution of the equation in the space with continuity of time. For this reason, we prove another form of solution of the equation in Fourier-Besov space. In chapter 5th, we consider the application of oscillatory integral in higher order damped wave equation. It is found that the performance of the estimator is different: in the low frequency part it is a thermal estimator, while in the high frequency part it is an oscillatory integral. Therefore, this chapter first divides into three parts to estimate the point state estimation of the kernel. Then we obtain the estimates of the fundamental solutions in L _ p space, and further use these estimates to obtain a global solution result of the equation.
【学位授予单位】：浙江大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O174.2

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