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

本赠款最显著的成果是确立了一个命题，即Navier-Stokes方程和相关方程中的非线性对流项可以防止解被炸毁。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，这些爆炸问题后来得到了统一的处理。这些方程证明了许多定理，但最重要的是：Proudman-Johnson方程的解在参数足够小的情况下不会爆破。Okamoto与C.H.Cho一起研究了一类允许爆破的半线性抛物方程的有限差分格式。研究了Nakagawa格式，证明了当真解光滑时，有限差分解收敛于真解，而且数值爆破时间收敛于真爆破时间。非线性波动方程的推广正在进行中。我们对IMT型的一维求积规则进行了显著的改进，并从数学上证明了在进行一定的参数调整时，其性能与DE规则一样好。松尾发明了离散变分方法，使我们能够推导出一种守恒量的有限差分/元方法，并将其应用于Camassa-Holm方程。

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.