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。对由Euler方程和Navier-Stokes方程控制的三维流的类别进行精确的数值计算。他们认为，这两种解决方案都会发展出诸如各种规范之类的奇异性，往往会在有限的时间内无限。实际上，这些实验提出的后来，康斯坦丁（Constantine）显示了欧拉方程情况的奇异性的发展。对于Navier-Stokes方程而言，尚未证明2。研究了Navier-Stokes方程的一些轴向对称性解决方案，发现该溶液具有内部层，因为雷诺数趋向于无穷大。特别是在适当的条件下，证明内部层存在3。当外部周期力对流体作用时，研究并阐明了它们的分叉。4。通过两种周期模式的相互作用，研究了薄层流体的一些周期性行进波，而正常形式的分析将复合模式的动力学完全分类为5。为了开发用于热对流问题解决方案空间的全球理论，尤其是在滚动型解决方案上应用计算机辅助证明，并证明了全球分叉分支的存在。空间。