Date: 14 April 2025 - 18 April 2025
Abstract:
The goal of these lectures is to present a rigorous mathematical analysis of the interaction of planar vortices in two-dimensional viscous flows at high Reynolds numbers. To keep the discussion as simple as possible, we focus on the particular case of a pair of vortices with equal or opposite circulations, in the well-prepared situation where these structures originate from point vortices at initial time. In the large Reynolds number regime, we construct an approximate solution of the 2D Navier-Stokes equations as a power series in the (time-dependent) aspect ratio of the viscous vortex pair. In particular, we compute the deformation of the stream lines due to vortex interactions, and we determine the leading order correction to the translation or rotation speed of the vortex centers due to finite size effects. We then show that the exact solution remains close to our approximation over a time interval that increases boundlessly as the viscosity parameter goes to zero. The proof relies on stability estimates derived from Arnold’s variational characterization of the steady states of the 2D Euler equation.