Mean-Field Convergence of Point Vortices to the Incompressible Euler Equation with Vorticity in $$L^\infty $$

Abstract

We consider the classical point vortex model in the mean-field scaling regime, in which the velocity field experienced by a single point vortex is proportional to the average of the velocity fields generated by the remaining point vortices. We show that if at some time the associated sequence of empirical measures converges in a renormalized $${\dot{H}}^{-1}$$ sense to a probability measure with density $$\omega ^0\in L^\infty ({\mathbb {R}}^2)$$ and having finite energy as the number of point vortices $$N\rightarrow \infty $$ , then the sequence converges in the weak-* topology for measures to the unique solution $$\omega $$ of the 2D incompressible Euler equation with initial datum $$\omega ^0$$ , locally uniformly in time. In contrast to previous results Schochet (Commun Pure Appl Math 49:911–965, 1996), Jabin and Wang (Invent Math 214:523–591, 2018), Serfaty (Duke Math J 169:2887–2935, 2020), our theorem requires no regularity assumptions on the limiting vorticity $$\omega $$ , is at the level of conservation laws for the 2D Euler equation, and provides a quantitative rate of convergence. Our proof is based on a combination of the modulated-energy method of Serfaty (J Am Math Soc 30:713–768, 2017) and a novel mollification argument. We contend that our result is a mean-field convergence analogue of the famous theorem of Yudovich (USSR Comput Math Math Phys 3:1407–1456, 1963) for global well-posedness of 2D Euler with vorticity in the scaling-critical function space $$L^\infty ({\mathbb {R}}^2)$$ .