本文關鍵詞：Fourier-Besov空間與振蕩積分及其應用　出處：《浙江大學》2015年博士論文　論文類型：學位論文

  更多相關文章： Fourier-Besov空間 乘積估計 廣義Navier-Stokes方程 振蕩積分 高階阻尼波動方程

【摘要】：調和分析中的方法技巧,系統(tǒng)化的概念或理論,都在方程的實際應用中,發(fā)揮巨大的作用.本文主要利用調和分析中的Littlewood-Paley理論綜合性的討論Fourier-Besov空間的定義、性質,并得到其在廣義Navier-Stokes方程中的一些應用；此外本文利用振蕩積分的理論方法研究了阻尼波動方程的一些估計及適定性.論文分為五個章節(jié),其主要內容安排為：第一章回顧本文課題的研究背景與現(xiàn)狀,綜合論述Fourier-Besov空間的發(fā)展和已有研究成果,廣義Navier-Stokes方程的研究情況以及阻尼波動方程的一些進展.并且在比較的基礎上,給出本文的主要定理.第二章集中討論Fourier-Besov空間.簡要回顧Littlewood-Paley理論和Besov空間的定義以及一些基本性質.通過與Besov空間類似的方式定義Fourier-Besov空間,從這一定義出發(fā),討論這個空間的等價形式、與其它空間的關系、包含嵌入、插值等性質.尤其是通過Bony分解的方法,給出Fourier-Besov空間的乘積估計.這一章的內容,也是在后面的兩章Fourier-Besov空間的應用中經常要用的.第三章第四章給出Fourier-Besov空間在廣義Navier-Stokes方程中的一些應用,綜合性的考慮廣義Navier-Stokes方程在Fourier-Besov空間中的性質.第三章考慮小初值的全局適定性,并在此基礎上證明解關于時間的全局衰減性.尤其是得到了方程在端點情形β=1/2時的一個全局適定性.第四章則研究廣義Navier-Stokes方程在Fourier-Besov空間中解的爆破準則以及空間正則性.為證明爆破準則關鍵在于構造方程在關于時間具有連續(xù)性的空間中的解,為此證明了方程在Fourier-Besov空間中另外一種形式的解,而空間正則性則采用Gevrey類的辦法.第五章考慮振蕩積分在高階阻尼波動方程中的應用.通過基本解的表達形式,發(fā)現(xiàn)其核算子的表現(xiàn)呈現(xiàn)著不同的變化：在低頻部分表現(xiàn)為熱核算子,而在高頻部分是一個振蕩積分的形式.因此這一章首先分成三部分估計核的點態(tài)估計,從而得到基本解在Lp空間上的估計,并進一步利用這些估計得到方程的一個全局解結果.
[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.
【學位授予單位】：浙江大學
【學位級別】：博士
【學位授予年份】：2015
【分類號】：O174.2

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本文編號：1413960