Researches on singularities arising in flows and waves

1.对一类由欧拉方程和Navier-Stokes方程控制的三维流动进行了精确的数值计算。他们认为，这两种解都发展出奇点，如各种范数在有限时间内趋于无穷大。事实上，从这些实验中可以看出，康斯坦丁后来证明了欧拉方程奇点的发展。对于纳维-斯托克斯方程，这一点还没有得到证明。研究了Navier-Stokes方程的一些轴对称相似解，发现当雷诺数趋于无穷大时，解具有内层。特别是在适当的条件下，证明了内层的存在。研究了外周期力作用于流体时的驻波、周期波、倍周期波和混沌波等各种波，并阐明了它们的分叉现象。研究了薄层流体在两个周期模相互作用下的一些周期行波，规范形分析对复模的动力学行为进行了完整的分类。为了发展热对流问题解空间的整体理论，利用计算机辅助证明了滚动型解的全局分支的存在性，并通过数值计算得到了三维情形下滚动型、矩形和六角型解的形态，并检验了它们在空间中的稳定性和分支.

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.