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

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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.