# Scaling-Critical Mean-Field Theory of Point Vortices and Other Coulomb Systems

# Scaling-Critical Mean-Field Theory of Point Vortices and Other Coulomb Systems

We consider classical Helmholtz-Kirchoff point vortices, a model example of a system with Coulomb interactions. In the mean-field regime where the magnitudes of the vortex circulations are inversely proportional to the number of vortices, which is very large, we expect the evolution to effectively be described by the vorticity formulation of the two-dimensional incompressible Euler equation. We will present a result on this approximation problem when the limiting vorticity is only in L∞, a scaling-critical function space for the well-posedness of the equation. We will also discuss the generalization of our result to higher-dimensional Coulomb systems and systems where multiplicative noise of transport-type is added to the dynamics. Time permitting, we will discuss some of the goals and challenges of going beyond mean-field theory.