  We consider a nonlinear model equation, known as the Localized Induction
Equation, describing the motion of a vortex filament immersed in an
incompressible and inviscid fluid. We show stability estimates for an
arc-shaped vortex filament, which is an exact solution to an initial-boundary
value problem for the Localized Induction Equation. An arc-shaped filament
travels along an axis at a constant speed without changing its shape, and is
oriented in such a way that the arc stays in a plane that is perpendicular to
the axis. We prove that an arc-shaped filament is stable in the Lyapunov sense
for general perturbations except in the axis-direction, for which the
perturbation can grow linearly in time. We also show that this estimate is
optimal. We then apply the obtained stability estimates to study the stability
of a circular vortex filament under some symmetry assumptions on the initial
perturbation. We do this by dividing the circular filament into arcs, apply the
stability estimate to each arc-shaped filament, and combine the estimates to
obtain estimates for the whole circle. The optimality of the stability
estimates for an arc-shaped filament also shows that a circular filament is not
stable in the Lyapunov sense, namely, certain perturbations can grow linearly
in time.
