The Vortex-Wave Equation with a Single Vortex as the Limit of the Euler Equation

In this article we consider the physical justification of the Vortex-Wave equation introduced by Marchioro and Pulvirenti (Mechanics, analysis and geometry: 200 years after Lagrange, North-Holland Delta Ser., Amsterdam, North-Holland, pp. 79-95, 1991), 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) a (c) L (a) 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.