Approximately half of my struggles have been to prove theorems about particular problems in fluid mechanics, the aim being to establish existence and properties of solutions in more detail than the general theory yields or for cases where the general theory does not apply. Examples of such inviscid flows (governed by the Euler equations) are steady vortex rings, the highest wave on water (about which Stokes conjectured in 1880) and the water entry of a wedge. Examples of such viscous flows (governed by the Navier-Stokes equations) are separating flow in channels, diffusing vortex rings and rows of diffusing vortices in the plane.

These problems and questions asked by colleagues have led me to a ragbag of topics in Analysis: singular perturbations, the effect of the bad boundary of a domain on spaces of functions defined on that domain, symmetry by way of the maximum principle in elliptic problems, the increase of electrostatic capacity with asymmetry and a little theory of the Fourier transform on a space of slowly decreasing functions.

This page was last modified on May 6, 2010 by Helen Higgins