Researches on singularities arising in flows and waves

流动和波浪中出现的奇点研究

基本信息

批准号：
11214204
负责人：
NISHIDA Takaaki
金额：
$ 12.8万
依托单位：
Kyoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research on Priority Areas (B)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11214204/
关键词：
Nonlinear Partial Differential Equation Navier-Stokes Equation Heat Convection Problem Thin Layer Flow Bifurcation Problem Interior Transition Layer Nonlinear Wave Computer Assisted Proof Boussinesq方程式特異性浅水波乱流 Faraday wave 渦およ

项目摘要

1. The precise numerical computation are performed for a class of three dimensional flows governed by the Euler Equation and Navier-Stokes equation. They suggest that both solutions develop the singularities such as the various norms tend to infinity in finite time. In fact suggested by these experiments later Constantine showed the development of the singularities for the case of the Euler equation. That for the Navier-Stokes equation is not yet proved.2. Some axially symmetric similarity solutions are investigated for the Navier-Stokes equation and it is found that the solution has an interior layer as the Reynolds number tends to infinity. Especially in appropriate conditions it is proved that the interior layer exists.3. When the external periodic forces act on fluids, various waves such as stationary, periodic, doubly periodic and chaotic waves are investigated and clarified the bifurcation of them.4. Some periodic traveling waves of the thin layer fluids are investigated on the modulations by the interaction of two periodic modes and the normal forms analysis classifies the dynamics of complex modes completely.5. To develop a global theory for the solution space of heat convection problem, a computer assisted proof is applied especially on the roll-type solutions the global bifurcation branch is obtained and proved about the existence.Pattern formations in the three dimensional case such as roll-type, rectangle-type and hexagonal-type solutions are obtained by the numerical computations and examined their stability and bifurcation branches globally in the space.

1.对欧拉方程和纳维-斯托克斯方程所控制的一类三维流动进行精确的数值计算。他们认为这两种解决方案都会产生奇点，例如各种范数在有限时间内趋于无穷大。事实上，后来康斯坦丁通过这些实验表明了欧拉方程情况下奇点的发展。 Navier-Stokes方程的这一点尚未得到证明。 2．研究了纳维-斯托克斯方程的一些轴对称相似解，发现随着雷诺数趋于无穷大，该解具有内层。特别是在适当的条件下，证明了内层的存在。 3．当外周期力作用于流体时，研究了稳态波、周期波、双周期波、混沌波等各种波，并阐明了它们的分岔。 4．研究了薄层流体的一些周期性行波对两种周期模态相互作用的调制作用，并通过范式分析对复杂模态的动力学进行了完整的分类。 5．为了发展热对流问题解空间的全局理论，计算机辅助证明特别适用于滚动型解，获得了全局分岔分支并证明了其存在性。三维情况下的模式形成，例如滚动通过数值计算得到了型、矩形型和六边形型解，并检验了它们在空间中全局的稳定性和分岔分支。