Numerical and mathematical analysis to low-dimensional solutions in. mathematical fluid problems

(1)考虑平面上用二维欧拉方程描述的点涡问题。(I)讨论了五点涡相对平衡的稳定性。我们发现，一些不稳定的组态表现出弛豫振荡。还发现了具有松弛振荡的四点涡和三点涡。给出了数学上的证明。(Ii)当涡旋相互靠近时，出现数值困难。为了克服这一困难，我们提出了一种数值格式。(2)我们考虑了球体中的点涡。当两个旋涡固定在球体的极点处时，证明了某些构型的稳定性和对中心流形的约化。(3)考虑了旋涡流动的三维线性不稳定性。利用带哈密顿量的谱理论，我们找到了某些情况下涡环不稳定的主要因素。(4)为了分析涡环在非线性区域的不稳定性，我们引入了一个弱非线性系统，用多尺度方法。利用该系统，我们发现了Windall失稳和失稳模式的机理。(5)利用奇异极限方法，提出并分析了经典单相Stefan问题和渗流问题的数值方法。(6)对于不混溶的两相流，我们考虑了一种无通量的有限元方法，该方法具有很好的质量守恒性。给出了数值解的误差估计和收敛结果。(7)将数值验证方法应用于二维流体问题中的驱动腔问题。我们成功地验证了一些稳定的解决方案。

(1) We consider problems of point vortices in a plane, which are described by the two-dimensional Euler equation. (i) Stability of relative equilibria for five point vortices is treated. We found that some unstable configurations exhibit the relaxation oscillation. Four and three point vortices which exhibit the relaxation oscillation are also found. Mathematical justifications are made. (ii) When vortices come close each other, numerical difficulty occurs. To overcome this difficulty, we propose a numerical scheme.(2) We consider the point vortices in a sphere. When two vortices are fixed at the poles of sphere, the stability of some configuration and reduction to a center manifold are shown.(3) Three-dimensional linear instability of vortex flow is considered. By using spectral theory with Hamiltonian, we found a primary factor of instability for some cases.(4) To analyze the instability of vortex ring in the nonlinear region, we introduce a weakly nonlinear system by multi-scale methods. By using this system, we found the mechanism of the Windall instability and unstable modes.(5) By singular limit methods, we propose and analyze the numerical methods to the classical one-phase Stefan problems and the flow through porous media.(6) For immiscible two-phase flow, we consider a flux-free finite element method, which shows good conservation of mass. We obtain error estimates and convergence of numerical solution.(7) We apply a numerical verification method to the driven cavity problem, which is one of famous two-dimensional fluid problems. We succeeded in the verification of some steady solutions.