Understand
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 suitable 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, 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 and a novel mollification argument.