Mathematical Open Problems of the Navier-Stokes Equations

对Navier-Stokes方程、Burgers方程和反应扩散方程的研究取得了进展。Okamoto和Shoji对表面波的分岔进行了数值实验。发现了新的分岔图，并将由世界科学公司以教科书的形式出版。Okamoto和Sakajo对二维和三维涡旋片运动进行了数值计算。池田和池田考虑了三个竞争物种的反应扩散方程组。它们阐明了稳态和行进脉冲的结构。它们的稳定性也是决定的。K.Ohkitani和M.Yamada考虑的是所谓的湍流壳模型。通过数值方法，他们计算了系统的李雅普诺夫数，并研究了这些数的尺度特性。当粘度趋于零时，他们推导出一个渐近公式。nakaki考虑了旋涡斑块和点涡的运动。他发现，如果斑块的大小足够小，斑块的运动与点涡的运动非常相似，如果漩涡斑块的大小足够大，漩涡斑块的运动就会有很大的不同。Y.Kimura考虑二维双曲曲面上点涡的运动。导出了它的哈密顿形式，并研究了不变量的代数性质。T.Nishida对热对流的主方程Boussinesq方程进行了数值计算。特别地，他从数值上得到了定常对流的平凡解的分岔。他运用数值验证技术，推导出新的准则。

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.