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

本文基于实数分析、傅里叶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数学物理中几种不同情况下对naveer -Stokes方程的应用。1)研究了一种不可压缩的黏性流体通过刚体，用Oseen方程进行数学描述。我们研究了Oseen半群在外域的衰减性质，并展示了通过刚体的全局时间稳定性Navier-Stokes流动。2)我们研究了扰动半空间中不可压缩的粘性流动，例如通过高层建筑的流动。研究了扰动半空间中Stokes方程解的最优衰减性质，证明了具有小初始数据的扰动半空间中Navier-Stokes方程解的全局时间唯一存在性。3)在键合域上用Neumann边界条件证明了Stokes方程解的极大正则性。我们使用了Weis和Denk-Hieber-Pruss对算子值傅里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，似乎适用于抛物线型线性演化方程。此外，我们证明了描述粘性不可压缩流体孤立体瞬态运动的Navier - Stokes方程自由边界问题具有任意初始数据的局部强解的时间唯一存在性和具有少量初始数据的整体强解的时间唯一存在性。这个问题已经被Galdi， Galdi和Silvestre在L_2框架下研究过了。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。主要的困难来自系数多项式增长的一阶微分系统，它不隶属于拉普拉斯算子。我们开发了一种新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