Abstract : As a toy model for the viscous interaction of planar vortices, we consider the solution of the two-dimensional Navier-Stokes equation with singular initial data corresponding to a pair of point vortices with opposite circulations. In the large Reynolds number regime, we construct an approximate solution which takes into account the deformation of the stream lines due to vortex interactions, as well as the corrections to the translation speed of the dipole due to finite size effects. Using energy estimates based on Arnold's variational characterization of equilibria for the Euler equation, we then show that our approximation remains valid over a very long time interval, if the viscosity is sufficiently small. This is a joint work with Michele Dolce (Lausanne), which relies on previous studies in collaboration with Vladimir Sverak (Minneapolis).

Keywords : Navier-Stokes equations; two-dimensional vortices; interaction