Uniqueness for the vortex-wave system when the vorticity is constant near the point vortex

TL;DR

This paper proves the uniqueness of solutions for the vortex-wave system with a single point vortex when the vorticity is initially constant near the vortex, using an Eulerian velocity-based approach.

Contribution

It introduces a novel Eulerian method to establish uniqueness in the vortex-wave system under specific initial conditions.

Findings

Proves uniqueness for the vortex-wave system with constant vorticity near the vortex.

Employs an Eulerian velocity formulation to achieve the result.

Extends understanding of vortex dynamics with point vortices.

Abstract

We prove uniqueness for the vortex-wave system with a single point vortex introduced by Marchioro and Pulvirenti in the case where the vorticity is initially constant near the point vortex. Our method relies on the Eulerian approach for this problem and in particular on the formulation in terms of the velocity.