Applied Analysis for Nonlinear Systems

非线性系统的应用分析

基本信息

批准号：
14340035
负责人：
NISHIDA Takaaki
金额：
$ 7.74万
依托单位：
Kyoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-14340035/
关键词：
Nonlinear partial differential equations Global structure of solution space Heat convection problems Dynamical systems Nonlinear waves Computer assisted proof Navier-Stokes equation Bifurcation problems 粘性衝撃波外部問題

项目摘要

(1)Heat Convection Pronlem : To extend the bifurcation curves obtained by the local bifurcation theory into the analytically unknown region in the solution space, to investigate the change of stability of the solution on the extended bifurcation curves and to know the global bifurcation structure, we use new computer assisted analysis for Boussinesq equation. Especially, we showed by computer assisted proofs the existence of extended bifurcation curves of the roll-type solutions. We also formulate a method to determine the point of secondary bifurcation on the extended bifurcation curves.(2)The cavity flows of Navier-Stokes equation are proved to exist by a revised numerical verification method for the higher Reynolds number. We reformulated the Newton method for the fixed point equation in the infinite dimensional space.(3)The blow-up of the solution of Navier-Stokes equation is proved to be characterized by the two components of vorticity, which means that its three components are not necessary to protect the blow-up.(4)Forced nonlinear wave equations are investigated by the Newton method in the infinite dimensional Banach space. The inverse operator of linearized equation at the approximate (constructed by computers) solution can be approximated in the norm by a pseudo diagonal operator.(5)In the Lorenz model equation we proved the existence of singularly degenerate heteroclinic cycle, which is an invariant set. We suppose that it will give the chaotic attractor by a perturbation.

(1)热对流问题：将局部分岔理论得到的分岔曲线扩展到解空间中解析未知区域，研究解在扩展分岔曲线上稳定性的变化，了解全局分岔结构，我们使用新的计算机辅助分析 Boussinesq 方程。特别是，我们通过计算机辅助证明证明了卷型解的扩展分岔曲线的存在性。我们还制定了确定扩展分岔曲线上二次分岔点的方法。(2)通过改进的较高雷诺数数值验证方法证明了Navier-Stokes方程的空腔流动的存在。我们重新表述了无穷维空间中不动点方程的牛顿法。(3)证明了纳维-斯托克斯方程解的爆炸可以用涡量的两个分量来表征，这意味着它的三个分量是(4)在无限维Banach空间中用牛顿法研究了受迫非线性波动方程。线性化方程在近似解（由计算机构造）处的逆算子可以用伪对角算子在范数上进行近似。(5)在Lorenz模型方程中我们证明了奇简并异宿循环的存在，它是一个不变集。我们假设它会给混沌吸引子带来扰动。