Mathematics Colloquium
Vladimir Sverak
University of Minnesota
Title: Vortex Rings, Vortex Filaments, and the Geometry of Incompressible Fluid Equations
Date: Friday, December 6th
Place and Time: Love 101, 3:05-3:55 pm
Abstract. Fluid mechanics involves multiple layers of "emergence": the Boltzmann equation arises from molecular dynamics, the classical fluid equations emerge from the Boltzmann equation, and objects like vortex rings and vortex filaments emerge from the fluid equations. The filaments have again their own equation, linked to the non-linear Schroedinger equation. The step from the fluid equations to the motion of vortex filaments is often called the Local Induction Approximation Conjecture and - except for very simple cases - remains elusive at the PDE analysis level. Here, we will focus on the simple case of the slightly viscous vortex rings. Their law of motion, first conjectured by Saffman (based on a more precise version of a classical computation by Kelvin) can be derived from the Navier-Stokes and Euler equations. Establishing some stability of the structures is the main challenge. The proofs use the Poisson structure of the equations of motion, which we will explain along with its relevance for the vortex dynamics. Based on joint work with Thierry Gallay.