Applications of the dynamical systems theory and the singularity theory to mathematical fluid mechanics

Navier-Stokes方程是流体力学的基本偏微分方程组。方程解的正则性问题特别著名，但方程还伴随着其他同样重要的问题。本研究的目的是为了更好地理解这些方程和相关的反应扩散方程。我们在(1)新的解的发现，特别是具有内层的近奇异解的发现，(2)行进水波的分叉分析和新的水波计算方法的发展方面取得了实质性进展。S.-C.Kim和Okamoto考虑了菱形周期的Navier-Stokes流，并发现了一个奇怪的大雷诺数的稳定性交换。Craik和Okamoto研究了一个源于Navier-Stokes方程的三维动力系统。对其解的渐近行为进行了分类：发现了一个二分法，并用周期解的存在来解释其原因。X.Chen和Okamoto证明了Proudman-Johnson方程在Dirichlet边界条件下不允许爆破，从而解决了一个长达十年的公开问题。Nagayama和Okamoto研究了Navier-Stokes方程的某些轴对称自相似解，并证明了它们对于大雷诺数具有内层。K.-I.Nakamura，H.Yagisita和Okamoto严格地证明了Navier-Stokes方程的自相似解在Burgers涡的框架下爆炸了。Okamoto证明了CRapper波在行进水波中产生的唯一性定理。

The Navier-Stokes equations are the fundamental partial differential equations of fluid mechanics. The regularity problem of the solutions of the equations is particularly renowned but the equations also accompany other equally important issues. The present research aims at better understanding of the equations and related reaction-diffusion equations. We have achieved substantial progresses in (1)discovery of solutions of new kind, particularly nearly singular solutions possessing internal layers, (2)bifurcation analysis for progressive water waves and a development of new computational methods for water waves. S.-C.Kim and Okamoto considered rhombic periodic Navier-Stokes flows, and discovered a curious stability-exchange at large Reynolds numbers. A.Craik and Okamoto studied a three dimensional dynamical system stemming from the Navier-Stokes equations. Asymptotic behavior of its solutions was classified : a dichotomy was discovered and the reason was explained by the existence of a periodic solution. X.Chen and Okamoto proved that the Proudman-Johnson equation did not admit a blow up under the Dirichlet boundary condition, which resolved an open problem for ten years.Nagayama and Okamoto studied certain axisymmetric self-similar solutions of the Navier-Stokes equations, and proved that they possessed internal-layers for large Reynolds numbers. K.-I.Nakamura, H.Yagisita and Okamoto rigorously proved that self-similar solutions of the Navier-Stokes equations blew up in the framework of the Burgers vortex. Okamoto proved a uniqueness theorem for Crapper's waves arising in the progressive water waves.