A Study of Blow-up Problems and Singular Perturbation Problems arisingin Mathematical Fluid Mechanics

数学流体力学中的爆炸问题和奇异摄动问题的研究

基本信息

批准号：
17204008
负责人：
OKAMOTO Hisashi
金额：
$ 14.39万
依托单位：
Kyoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (A)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2007
项目状态：
已结题

项目摘要

The most notable outcome of the present grant-in-aid is to have established a proposition that a nonlinear convection term in the Navier-Stokes equations and related equations can prevent solutions from blowing up. We also explored De Gregorio's equation and the Proudman-Johnson equation to find that they are rich sources of blow-up problems. Besides, those blow-up problems turned out to be treated in a unified way. Many theorem were proved for these equations, but the following is the most significant : solutions of the Proudman-Johnson equation do not blow up if the parameter in it is small enough.Together with C.-H. Cho, Okamoto investigated finite difference schemes for a semi-linear parabolic equation which admits blow-up. Schemes of Nakagawa type were studied and we proved not only that the finite difference solution converges to the true solution while the true solution is smooth, but also that the numerical blow-up time converges to the true blow-up time. A generalization to nonlinear wave equations is in progress.Ooura made a significant improvement on one-dimensional quadrature rule of IMT type and proved mathematically that their performance is as good as DE rule if a certain parameter tuning is made.Matsuo invented discrete variational method, which enables us to derive a finite difference/element method preserving conservation quantities, and applied it to the Camassa-Holm equation.

当前赠款中最值得注意的结果是建立一个主张，即Navier-Stokes方程和相关方程中的非线性对流术语可以防止解决方案爆炸。我们还探索了De Gregorio的方程式和Proudman-Johnson方程，发现它们是爆炸问题的丰富来源。此外，这些爆炸问题被证明是以统一的方式进行的。这些方程式证明了许多定理，但是以下是最重要的：如果其中的参数足够小。 Cho，Okamoto调查了一个半线性抛物线方程的有限差方方案，该方程接受了爆炸。研究了中川类型的方案，我们不仅证明了有限差解决方案在真实解决方案时会收敛到真实解决方案，而且还证明了数值爆破的时间会收敛到真实的爆破时间。 A generalization to nonlinear wave equations is in progress.Ooura made a significant improvement on one-dimensional quadrature rule of IMT type and proved mathematically that their performance is as good as DE rule if a certain parameter tuning is made.Matsuo invented discrete variational method, which enables us to derive a finite difference/element method preserving conservation quantities, and applied it to the Camassa-Holm equation.