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

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