Littlewood-Paley理论及其在流体动力学方程中的应用

【作者】 苑佳；

【导师】 苗长兴；

【作者基本信息】 中国工程物理研究院 ， 应用数学， 2008， 博士

【摘要】 80年代以来,Fourier分析方法越来越多地被应用在线性和非线性偏微分方程的研究中.特别地,Littlewood-Paley分解和Bony仿积分解被证明是很有效的工具.Littlewood-Paley分解早在50多年前就被提出,最近才在PDE的框架下被系统地使用;而仿积分解是J.-M.Bony在1981年研究非线性双曲方程微局部奇性传播时提出的(参见[7]).本文致力于应用Littlewood-Paley分解和仿积分解的相关理论来处理Navier-Stokes方程,磁微极流体方程,广义MHD方程,耗散准地转方程和广义Camassa-Holm方程,研究这几种流体力学方程的基本性质,例如:解的存在性,弱解的正则性和强解的爆破准则.首先,第二章介绍Littlewood-Paley理论.具体的来说:我们介绍频率空间的局部化技术(也称为单位二进制分解或Littlewood-Paley分解)和Bony的仿积分解技术,应用Littlewood-Paley理论来刻画齐次和非齐次Besov空间,并列举出Besov空间的一些性质和估计.第三章研究Navier-Stokes方程,Navier-Stokes方程是刻画流体运动的基本方程,其整体适定性问题是当今数学界最关注的公开问题之一.而且二维Navier-Stokes方程相应的问题已经彻底解决.我们的目标是对n维Navier-Stokes方程(n≥3)建立强解的频谱层次上的爆破准则和弱解的正则性.第四章利用Bony的仿积分解技术,频段层次上的交换子估计,高-低频分解等调和分析方法对对三维磁微极流体方程的耗散项与非线性项进行细致的分析,建立一系列在混合时空空间上更为精细的估计,得到方程光滑解的适定性以及建立在频谱层次上,只依赖于速度场的涡度的Blow-up准则.第五章利用Littlewood-Paley分解和Bony的仿积分解技术,研究一类广义MHD方程的适定性及弱解的正则性、改进了一些已有的正则性结果.第六章利用一个由二进制分解和Bony的仿积分解建立成的交换子估计,简单而巧妙地获得了二维耗散准地转方程的一个正则性判别准则,改进了Dong和Chen[43]的结果.第七章研究Camassa-Holm方程,利用事实:Camassa-Holm方程经过简单的变换后在结构上与输运方程存在相似性,再使用Besov空间框架下输运方程的估计,紧性讨论以及Littlewood-Paley理论,得到广义Camassa-Holm方程的局部适定性.更多还原

【Abstract】 Since the 80’s,Fourier analysis methods have known a growing interest in the study of linear and nonlinear PDE’s.In particular,techniques based on Littlewood-Paley decomposition and paradifferential calculus have been proved to be very efficient.Littlewood-Paley decomposition has been introduced more than fifty years ago in harmonic analysis but its systematic use in the PDE’s framework is rather recent.Paradifferential calculus,as for it,has been introduced in 1981 by J.-M.Bony for the study of the propagation of microlocal singularities in nonlinear hyperbolic PDE’s(see[7]).In this thesis,we use theory of Littlewood-Paley decomposition and Bony’s paraproduct decomposition to study several types of hydrodynamic equations: Navier-Stokes equations,Magneto-Micropolar fluid equations,generalized Magnetohydrodynamics equations,dissipative Quasi-Geostrophic equations and generalized Camassa-Holm equations.We study their basic properties such as the local existence,the regularity of the weak solutions and the blow-up criteria of strong solutions etc.In the second chapter,we recall the theory of Littlewood-Paley.We introduce the Littlewood-Paley decomposition,by which we define Besov space and time-space Besov space.Then we introduce Bony’s paraproduct decomposition technique.Some properties and estimates of Besov space and Bony’s paraproduct are listed.In the third chapter,we study Navier-Stokes equations.Navier-Stokes equations are the basic equations describing the movement of the fluid,the problem of global existence is one of the open problem attracting many mathematicians.For the two dimensional case,corresponding problems have been solved.We consider n-dimension(n≥3) Navier-Stokes equations and set up the blow-up criteria of strong solutions and the regularity of the weak solutions.In the fourth chapter,by using of some harmonic methods including Bony’s paraproduct decomposition,commutator estimate based on frequency localization, high-low frequency decomposition,we impose detailed analysis on the dissipative term and the nonlinear term of the three dimensional Magneto-Micropolar fluid equations,set up a series of more refined estimates,we then obtain the wellposedness of the smooth solution.In the fifth chapter,using Littlewood-Paley decomposition and paradifferential calculus,we study local well-posedness and the regularity of the weak solutions of the generalized Magneto-hydrodynamics equations,refined some results.In the sixth chapter,we consider the 2D dissipative quasi-geostrophic equations and study the regularity criterion of the solutions.By means of a commutator estimate based on frequency localization and Bony’s paraproduct decomposition, we obtain a regularity criterion which improves the result of Dong and Chen[43].In the last chapter,we know that,after simple transform,Camassa-Holm equations have many similarity with transport equations.Using estimates of transform equations in the frame of Besov spaces,compactness arguments and Littilewood-Paley theory,we obtain local wellposedness of generalized CamassaHolm equations.更多还原