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Research of the Navier-Stokes equations by using the theory of Fourier analysis and semigroup theory

利用傅立叶分析和半群理论研究纳维-斯托克斯方程

基本信息

批准号：
11640156
负责人：
YAMAZAKI Masao
金额：
$ 1.47万
依托单位：
Hitotsubashi University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640156/
关键词：
Navier-Stokes equations Lorentz spaces Morrey spaces Weak-* convergence periodic solutions almost periodic solutions exterior domains stability *-弱収束 Besov空間 Hardy空間

项目摘要

We studied the Navier-Stokes equation on the whole space or on exterior domains. We have already studied the unique existence and the stability under initieal perturbations of stationary solutions with external forces independent of time. In this research we considered the case where external force depends on the time-variable, and studied the unique existence and the stability of solutions. This research generalizes similar researches on time-periodic solutions and solutions almost periodic in time.On the whole space we employ the Morrey spaces as the space of solutions, and succeeded in generalizing the results for usual L^p-spaces obtained by Professors Hideo Kozono, Mitsuhiro Nakao and Yasushi Taniuchi. On exterior domains we employ the weak-L^p spaces as the space of solutions, and we succeeded in relaxing the assumptions on the external forces very much, and firstly obtained conditions sufficient for the unique existence of solutions in 3-dimensional exterior domains.The results for the Morrey spaces can be obtained in a manner similar to that employed in our previous study. On the other hand, in the proof of the results for weak-L^p spaces, it is essential to show that the integral of functions with values in a Banach space converges, where the integral is considered to diverge in general. In order to show this fact, we first consider the family of the Lorentz spaces which generalizes the weak-L^pA spaces, and we improved estimates of L^p-L^q type, which is often employed in previous studies, by using real interpolation, and the we used the duality property between the Lorentz spaces.

我们研究了全空间和外部区域上的Navier-Stokes方程。我们已经研究了具有与时间无关的外力的定态解的唯一存在性和在初始扰动下的稳定性。在本研究中，我们考虑了外力依赖于时间变量的情形，研究了解的存在唯一性和稳定性。本文推广了关于时间周期解和时间概周期解的类似研究，在整个空间上，我们采用Morrey空间作为解的空间，并成功地推广了Hideo Kozono，Mitsuhiro Nakao和Yasushi Taniuchi教授关于一般L^p-空间的结果。在外部区域上，我们采用弱L ^p空间作为解的空间，大大放宽了对外力的假设，首次得到了三维外部区域上解存在唯一性的充分条件，在Morrey空间上得到的结果与我们以前的研究类似.另一方面，在证明弱L ^p空间的结果时，必须证明值在Banach空间中的函数的积分收敛，而积分一般被认为是发散的。为了证明这一点，我们首先考虑了Lorentz空间族，它是弱L ^pA空间的推广，利用真实的插值和Lorentz空间之间的对偶性质，改进了以往研究中常用的L^p-L^q型估计.