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We consider the classical point vortex model in the mean-field scalingregime, in which the velocity field experienced by a single point vortex isproportional to the average of the velocity fields generated by the remainingpoint vortices. We show that if at some time, the associated sequence ofempirical measures converges in a suitable sense to a probability measure withdensity $\omega^0\in L^\infty(\mathbb{R}^2)$ and having finite energy, as thenumber of point vortices $N\rightarrow\infty$, then the sequence converges inthe weak-* topology for measures to the unique solution $\omega$ of the 2Dincompressible Euler equation with initial datum $\omega^0$, locally uniformlyin time. In contrast to previous results, our theorem requires no regularityassumptions on the limiting vorticity $\omega$, is at the level of conservationlaws for the 2D Euler equation, and provides a quantitative rate ofconvergence. Our proof is based on a combination of the modulated-energy methodof Serfaty and a novel mollification argument. We contend that our result is amean-field convergence analogue of the famous theorem of Yudovich for globalwell-posedness of 2D Euler in $L^1(\mathbb{R}^2)\cap L^\infty(\mathbb{R}^2)$.

Mean-Field Convergence of Point Vortices without Regularity