On the dynamics of vortices in viscous 2D flows

TL;DR

This paper analyzes the evolution of vorticity in 2D viscous flows, providing quantitative estimates on how initial vortex concentrations evolve and interact, extending classical results to more general initial conditions and viscosity-independent estimates.

Contribution

It extends previous work by deriving uniform estimates for vortex dynamics with initial vorticity in L^p, p>2, and weak concentration, independent of viscosity.

Findings

Quantitative estimates on vortex concentration propagation

Extension to initial vorticity in L^p with p>2

Uniform estimates with respect to viscosity

Abstract

We study the 2D Navier--Stokes solution starting from an initial vorticity mildly concentrated near $N$ distinct points in the plane. We prove quantitative estimates on the propagation of concentration near a system of interacting point vortices introduced by Helmholtz and Kirchhoff. Our work extends the previous results in the literature in three ways: The initial vorticity is concentrated in a weak (Wasserstein) sense, it is merely $L^p$ integrable for some $p>2$, and the estimates we derive are uniform with respect to the viscosity.