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

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(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.