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

本文以实分析、傅立叶分析和泛函分析的最新发展为基础，研究了Stokes方程的谱分析及其在数学物理中几种不同情况下对Naveir-Stokes方程的应用。1)我们研究了不可压缩的粘性绕过刚体的流动，其数学描述为Osee方程。我们研究了Oseen半群在外部区域的衰减性质，证明了流经刚体的整体时间稳定性。2)我们研究了扰动半空间中的不可压缩粘性流动，例如描述了高层建筑的流动。研究了扰动半空间中Stokes方程解的最优衰减性质，证明了小初值扰动半空间中N-S方程解在时间上的整体唯一性。3)证明了带有Neumann边界…的Stokes方程解的最大正则性绑定域中的条件更多。我们使用Weis和Denk-Hieber-Pruss对算子值傅立叶分析的一些最新发展。与前人的结果相比，我们的方法非常简单，而且似乎适用于抛物型线性方程。此外，我们还证明了描述粘性不可压缩流体孤立体积瞬变运动的Navier-Stokes方程自由边界问题的强解在时间上的局部唯一性和初值不定时强解的全局唯一性。4)研究了旋转刚体不可压缩粘性流动。这个问题已经由Galdi和Galdi以及Silvestre在L_2框架内进行了研究。我们的主要贡献是证明了与线性化问题相关的连续半群的衰减估计。主要的困难来自一阶具有多项式增长系数的微分系统，该系统不隶属于拉普拉斯方程。我们开发了新的技术来研究光谱的高频部分。较少

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