Researches on singularities arising in flows and waves

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1. The precise numerical computation are performed for a class of three dimensional flows governed by the Euler Equation and Navier-Stokes equation. They suggest that both solutions develop the singularities such as the various norms tend to infinity in finite time. In fact suggested by these experiments later Constantine showed the development of the singularities for the case of the Euler equation. That for the Navier-Stokes equation is not yet proved.2. Some axially symmetric similarity solutions are investigated for the Navier-Stokes equation and it is found that the solution has an interior layer as the Reynolds number tends to infinity. Especially in appropriate conditions it is proved that the interior layer exists.3. When the external periodic forces act on fluids, various waves such as stationary, periodic, doubly periodic and chaotic waves are investigated and clarified the bifurcation of them.4. Some periodic traveling waves of the thin layer fluids are investigated on the modulations by the interaction of two periodic modes and the normal forms analysis classifies the dynamics of complex modes completely.5. To develop a global theory for the solution space of heat convection problem, a computer assisted proof is applied especially on the roll-type solutions the global bifurcation branch is obtained and proved about the existence.Pattern formations in the three dimensional case such as roll-type, rectangle-type and hexagonal-type solutions are obtained by the numerical computations and examined their stability and bifurcation branches globally in the space.