Cursillo “On vortex dynamics in two-dimensional or three-dimensional incompressible flows”,

por Evelyne Miot (CNRS and Ecole Polytechnique, investigadora invitada CMM)

– Lecture 1 (Lunes 30 Marzo, 16h-17:30h, Sala Seminario 5to. piso):
Vortices in incompressible fluids.

In this first lecture we will consider the Euler equations governing the motion of incompressible fluids, in particular in a two-dimensional setting. We will focus on the vortex solutions and present Marchioro and Pulvirenti’s result on the derivation of the point vortex system from the 2D Euler equations. We will also briefly mention the vortex dynamics in other related equations (the Navier-Stokes equation and the Gross-Pitaevskii equation).

– Lecture 2 (Miércoles 1 Abril, 12h-13:30h, Sala Seminario 5to. piso)
Convergence of the point vortex system to the 2D Euler equation.

This lecture will explore more in detail the connection between the discrete model, described by the point vortex system, and the continuous fluid dynamics given by the Euler equation. We will show how the results by Goodman, Lowengrub and Hou and Schochet ensure that the point vortex system is a good approximation of the Euler equation when the number of vortices is large.

– Lecture 3 (Jueves 2 Abril, 14:30h-16h, Sala Seminario 7mo. piso):
Vortex filaments.

We will study the analogous notion of point vortices in three dimensions, namely the vortex filaments. We will explain the formal derivation leading to the binormal curvature flow equation governing the motion of one single vortex filament. We will also relate the binormal curvature flow equation and the cubic 1D Schrödinger equation via the Hasimoto transform. Finally we will present a system of simplified equations proposed by Klein, Majda and Damodaran to describe the interaction of several almost parallel vortex filaments.