On the initial-boundary value problem for a vortex filament in an external flow under the localized induction approximation

Abstract

A vector equation describing the motion of a vortex filament in an external flow under the localized induction approximation is considered. The initial-boundary value problem for it is proved to have at least a solution when both ends of the filament are fixed. Unlike the case without an external flow, the stretch and shrink of the filament generally occur. But, for our solution, they are finite at any finite time.