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）热对流的前置：将通过局部分叉理论获得的分叉曲线扩展到解决方案空间中的分析未知区域，以研究解决方案在扩展分叉曲线上的稳定性的变化，并了解全球分叉结构，我们对BOUSSINESENESSESENESSESESESESESESESESESESESENESSINESSINESSINESSINESENESSESINESSINESSINESSINESSINESSINESSINESSINESSINESSINESSINESSINESSINESSINESSINESSINESSINESSINESSESESESESEQEATION。尤其是，我们通过计算机辅助证明了卷型溶液的扩展分叉曲线的存在。我们还制定了一种方法，以确定扩展分叉曲线上的次级分叉点。（2）通过修订后的雷诺数数量的修订后数值验证方法证明，Navier-Stokes方程的腔流得到证明存在。我们对无限尺寸空间中的固定点方程进行了重新制定。（3）事实证明，Navier-Stokes方程解决方案的爆炸以涡度的两个组成部分的特征，这意味着它的三个组成部分是不需要保护爆炸的三个组成部分。在近似（由计算机构建的）解决方案构造的线性化方程的逆操作员可以通过伪对角线算子近似。（5）在Lorenz模型方程中，我们证明了存在奇异退化的杂音循环的存在，这是一个不变的集合。我们认为它将通过扰动给予混乱的吸引子。