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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.对一类由Euler方程和Navier-Stokes方程控制的三维流动进行了精确的数值计算。他们认为，这两种解决方案发展的奇异性，如各种规范往往在有限时间内无穷大。事实上，建议这些实验后来君士坦丁表明发展的奇异性的情况下，欧拉方程。对于Navier-Stokes方程，这一点尚未得到证明.研究了Navier-Stokes方程的轴对称相似解，发现当雷诺数趋于无穷大时，解中存在内层。特别是在适当的条件下，证明了内层的存在.研究了流体在周期性外力作用下产生的各种波，如稳态波、周期波、双周期波和混沌波，并阐明了它们的分岔现象。研究了薄层流体中的一些周期行波对两个周期模相互作用的调制，规范形分析对复模动力学进行了完整的分类.为了发展热对流问题解空间的整体理论，应用计算机辅助证明方法，特别是对Roll型解，得到了其整体分支分支，并证明了分支的存在性.通过数值计算，得到了Roll型、矩形型和六边形型解在三维空间中的斑图形式，并研究了它们在空间中的整体稳定性和分支.