The equations which describe the motion of a vortex filament with or without an axial flow inside its core are considered. The initial and the initial-boundary value problems are proved to have unique and smooth solutions globally in time. These results are obtained by adding vanishing parabolic terms which conserve the length of the filament.Key words, vortex filament, perfect fluid, localized induction equation, initial and initialboundary value problems, unique and smooth solvability AMS subject classifications. 35Q, 76C 1. Introduction. The system of equations 1, on (1.2) for s R. One of our aims in this paper is to establish the unique and smooth solvability of the initial value problem (1.2) with (1.3) in the space where the curvature of the vortex filament ]v] tends to .zero as s , on the time interval [0, T] with any T > 0. In order to achieve it, we first investigate the parabolic regularization for > 0. After that, we let e -0.

It is known that the motion of a vortex filament with axial flow in a perfect fluid is approximately described by a generalization of the localized induction equation. The unique solvability of the initial value problem for it is first established by parabolic regularization. (1.2)x t = x s x x ss + a{x sss 4-(3/2)* ss x (x s x x ss )}, when the vortex filament has an axial flow within its thin vortex core. Here a is a real constant representing the magnitude of the axial flow effect.

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Abstract. A sequence of solutions to the Galerkin approximation of a nonstationary magnetohydrodynamic system is proved to converge to a measure-valued solution, in the sense of R. J. DiPerna-A. J. Majda, to the three-dimensional stationary Euler equations.

The Galerkin approximation of the nonstationary system of G. K. Vallis et al. is proved to yield generalized solutions to the stationary Euler equations in the axisymmetric case. This result is obtained by letting the number of basis functions and the time tend to infinity simultaneously.