The Vortex-Wave equation with a single vortex as the limit of the Euler equation

TL;DR

This paper rigorously justifies the Vortex-Wave equation as a limit of the Euler equation for a single vortex with concentrated vorticity, demonstrating convergence of solutions under specific initial conditions.

Contribution

It provides a mathematical proof that solutions of the Euler equation converge to a weak solution of the Vortex-Wave equation for a single vortex scenario.

Findings

Convergence of Euler solutions to the Vortex-Wave equation in velocity.

Introduction of a weak solution concept for the Vortex-Wave equation.

Validation of the Vortex-Wave model as a limit of Euler dynamics.

Abstract

In this article we consider the physical justification of the Vortex-Wave equation introduced by Marchioro and Pulvirenti in the case of a single point vortex moving in an ambient vorticity. We consider a sequence of solutions for the Euler equation in the plane corresponding to initial data consisting of an ambient vorticity in $L^1\cap L^\infty$ and a sequence of concentrated blobs which approach the Dirac distribution. We introduce a notion of a weak solution of the Vortex-Wave equation in terms of velocity (or primitive variables) and then show, for a subsequence of the blobs, the solutions of the Euler equation converge in velocity to a weak solution of the Vortex-Wave equation.