Nonlinear Partial Differential Equations

Date: December 4, 2018

Time: 3:00PM - 4:00PM

Location: BLOC 628

Speaker: Hakima Bessaih, University of Wyoming

Title: Nonlinear PDE's Seminar

Abstract:

Date: Tuesday, December 4, 2018 (please notice this is different date than what we have announced earlier)

Title: Mean field limit of interacting filaments for 3d Euler equation

Abstract: The 3D Euler equation, precisely local smooth solutions of class $H^s$ with $s>5/2$ are obtained as a mean field limit of finite families of interacting curves, the so called vortex filaments, described by means of the concept of 1-currents. This is achieved by first replacing the true Euler equation by a mollified one through the regularization of the Biot-Savart law through a small coefficient $\epsilon$. Families of N interacting curves are considered, with long range mean field type interaction, that depends on the coefficient $\epsilon$. When $N$ goes to infinity, the limit PDE is vector-valued (mollified Euler equation) and each curve interacts with a mean field solution of the PDE.

This target is reached by a careful formulation of curves and weak solutions of the PDE which makes use of 1-currents and their topologies. The main results are based on the analysis of a nonlinear Lagrangian-type flow equation. Most of the results are deterministic; as a by-product, when the initial conditions are given by families of independent random curves, we prove a propagation of chaos result.