Mathematical Open Problems of the Navier-Stokes Equations

在Navier-Stokes方程、Burgers方程和反应扩散方程的研究方面取得了进展。Okamoto和Shoji对表面波的分叉进行了数值实验。世界科学公司发现了新的分岔图，并将以教科书的形式出版。Okamoto和Sakajo对二维和三维涡片运动进行了数值计算。T.Ikeda和H.Ikeda考虑了三个相互竞争的物种的特定反应扩散方程组。它们阐明了稳态脉冲和行进脉冲的结构。他们的稳定性也是确定的。K.Ohkitani和M.Yamada考虑了被称为湍流的壳模型。用数值方法计算了系统的Lyapunov数，并研究了这些数的标度性质。当粘性趋于零时，他们推导出一个渐近公式。T.Nakaki考虑涡片的运动以及点涡旋的运动。他发现，如果斑块的大小足够小，那么斑块的运动与点涡旋的运动非常相似，而如果斑块的大小很大，涡斑的运动就会有很大的不同。Y.Kimura考虑了二维双曲曲面上点涡的运动。推导了它的哈密顿形式，并研究了不变量的代数性质。T.Nishida数值计算了Boussinesq方程，这是热对流的主方程。特别地，他在数值上得到了定常对流平凡解的分叉。他应用了数字验证技术，并得出了新的标准。

Progress is made in the study of the Navier-Stokes equation, the Burgers equations, and the reaction-diffusion equations. Okamoto and Shoji performed numerical experiments on the bifurcation of surface waves. New bifurcation diagrams are found and will be published in a form of textbook by World Scientific Inc. Okamoto and Sakajo compute numerically two-dimensional and three-dimensional vortex sheet motion. T.Ikeda and H.Ikeda consider a certain system of reaction-diffusion equations for three competing species. They clarify the structure of steady-states and traveling pulses. Their stability is also determined. K.Ohkitani and M.Yamada consider what is called the shell model of the turbulence. By numerical methods, they compute the Lyapunov numbers of the system and they study the scaling properties of the numbers. They derive an asymptotic formula as the viscosity tends to zero. T.Nakaki consider the motion of vortex patches as well as point vortices. He finds that the motion of the patches are quite similar to that of point vortices if the size of the patches are small enough and that the motion of vortex patches are substantially different if the sizes are large. Y.Kimura considers the motion of point vortices on two-dimensional hyperbolic surfaces. Its Hamiltonian formalism are derived and the algebraic properties of the invariants are studied. T.Nishida numerically computes the Boussinesq equations, which are the master equations for the thermal convection. In particular he obtains numerically the bifurcation from the trivial solution to the stationary convective flow. He applies the numerical verification technique and derives new criteria.