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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.

本补助金最显著的成果是确立了一个命题，即纳维尔-斯托克斯方程和相关方程中的非线性对流项可以防止解爆炸。我们还研究了De Gregorio方程和Proudman-Johnson方程，发现它们是爆破问题的丰富来源。此外，那些爆炸性的问题原来是在一个统一的方式处理。对于这类方程，人们证明了许多定理，但其中最重要的是：如果Proudman-Johnson方程中的参数足够小，则方程的解不会爆破。H. Cho，Okamoto研究了一类半线性抛物型方程的有限差分格式。研究了Nakagawa型格式，证明了有限差分解收敛于真解，而真解是光滑的，而且数值爆破时间收敛于真爆破时间。对非线性波动方程的推广正在进行中，Ooura对IMT型的一维求积规则作了重大改进，并在数学上证明了如果进行一定的参数调整，它们的性能与DE规则一样好; Matsuo发明了离散变分方法，使我们能够导出一种保持守恒量的有限差分/元方法，并将其应用于Camassa-Holm方程。