On A Singular Initial-Value Problem For The Navier–Stokes Equations

This paper presents a recent result for the problem introduced eleven years ago by Fraenkel and McLeod [A diffusing vortex circle in a viscous fluid. In IUTAM Symposium on Asymptotics, Singularities and Homogenisation in Problems of Mechanics, Kluwer (2003), 489–500], but described only briefly there. We shall prove the following, as far as space allows. The vorticity (Formula presented.) of a diffusing vortex circle in a viscous fluid has, for small values of a non-dimensional time, a second approximation (Formula presented.) that, although formulated for a fixed, finite Reynolds number (Formula presented.) and exact for (Formula presented.) (then (Formula presented.)), tends to a smooth limiting function as (Formula presented.). In §§1 and 2 the necessary background and apparatus are described; §3 outlines the new result and its proof.