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

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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.