In this talk, we consider the motion of a vortex ring under the influence of external flow. This can be seen as an idealization of the motion of a bubble ring traveling through water, where environmental flow is also present.
The motion is described as an initial value problem posed on the one-dimensional torus for a closed vortex filament. The equation of motion is a nonlinear dispersive type equation and is an extension of the Localized Induction Equation (LIE). The LIE is one of the most oldest and fundamental model equation describing the motion of a vortex filament, and the equation we consider in this talk is a generalization of the LIE which takes into account the presence of external flow. The time-local solvability of the initial value problem will be presented, focusing on the derivation of energy estimates needed in order to prove the solvability.