Study of the stability of solutions to some nonlinear evolution equations based on recent development of real analysis

本文基于真实的分析、Fourier分析和泛函分析的最新发展，研究了Stokes方程的谱分析及其在Naveir-Stokes方程中几种不同情况下的应用。我们研究了Oseen半群在外部区域的衰减性质，并证明了绕刚体的Navier-Stokes流的全局时间稳定性。2）我们研究了扰动半空间中的不可压粘性流，例如描述绕高层建筑物的流。我们研究了扰动半空间中Stokes方程解的最优衰减性质，证明了扰动半空间中Navier-Stokes方程解在小初值下的整体时间唯一存在性; 3）证明了具有Neumann边界的Stokes方程解的极大正则性 ...更多信息 在一个绑定域的条件。我们使用Weis和Denk-Hieber-Pruss的算子值傅立叶分析的一些最新发展。与以往的结果相比，我们的方法非常简单，而且似乎适用于抛物型线性演化方程。此外，我们还证明了描述粘性不可压缩流体孤立体瞬态运动的Navier-Stokes方程自由边界问题在任意初值条件下强解的局部时间唯一性和在小初值条件下强解的整体时间唯一性。Galdi和Galdi和Silvestre在L_2框架下已经研究了这个问题。我们的主要贡献是给出了与线性化问题相关的连续半群的衰减估计。主要困难来自于一阶多项式增长系数微分系统，它不服从拉普拉斯算子。我们开发了新的技术来研究频谱的高频部分。少

We study the spectral analysis of Stokes equations based on the recent development of the real analysis, Fourier analysis and functional analysis and its application to the Naveir-Stokes equations in several different situations arising from the mathematical physics.1) We studied an inncompressible viscous flow past a rigid body, which is mathematically described by Oseen equations. We studied the decay properties of the Oseen semigroup in the exterior domain and showed global in time stability Navier-Stokes flow past a rigid body2) We studied an inncompressible viscous flow in a perturbed half-space which describes for example flow past high buildings. We studied an optimal decay properties of solutions to the Stokes equations in a perturbed half-space and proved a global in time unique existence of solutions to the Navier-Stokes equations in a perturbed half-space with small initial data.3) We proved the maximal regularity of solutions to the Stokes equation with the Neumann boundry … More condition in a bonded domain. We use some recent development of the operator-valued Fourier analysis by Weis and Denk-Hieber-Pruss. Our method is very simple compared with previous results and seems to be applicable to linear evoulution equations of parabolic type. Moreover, we proved local in time unique existence of strong solutions with arbitrary initial data and global in time unique existence of strong solutions with some small initial data of the free boundary problem of Navier Stokes equations which describes the transient motion of an isolated volume of viscous incompressible fluid4) We studied an inncompressible viscous flow past a rotating rigid body. This problem was already studied by Galdi and Galdi and Silvestre in the L_2 framework. Our main contribution is to show the decay estimate of the continuous semigroup associated with linearized problem. The main difficulty comes from first order differential system with polynomially growing coefficients which is not subordinated by the Laplacian. We developed new technique to investigate the high frequency part of the spectrum. Less