论文标题
没有规律性的点涡流的平均场融合
Mean-Field Convergence of Point Vortices without Regularity
论文作者
论文摘要
我们考虑在平均场缩放机制中的经典涡流模型，其中单点涡流所经历的速度场与其余点涡流产生的速度场的平均值成正比。 We show that if at some time, the associated sequence of empirical measures converges in a suitable sense to a probability measure with density $ω^0\in L^\infty(\mathbb{R}^2)$ 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 $ω$ of the 2D不可压缩的Euler方程，具有初始基准$ω^0 $，及时均匀。与以前的结果相反，我们的定理不需要关于限制涡度$ω$的规律性假设，在2D Euler方程的保护法处，并且提供了收敛的定量率。我们的证明是基于调制能量的奴隶制方法和新颖的弹性论点的组合。我们认为，我们的结果是Yudovich著名定理的平均场融合类似物，以$ l^1（\ Mathbb {r}^2）\ Cap l^\ infty（\ Mathbb {r Mathbb {r}^2）$。
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 $ω^0\in L^\infty(\mathbb{R}^2)$ 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 $ω$ of the 2D incompressible Euler equation with initial datum $ω^0$, locally uniformly in time. In contrast to previous results, our theorem requires no regularity assumptions on the limiting vorticity $ω$, 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. We contend that our result is a mean-field convergence analogue of the famous theorem of Yudovich for global well-posedness of 2D Euler in $L^1(\mathbb{R}^2)\cap L^\infty(\mathbb{R}^2)$.
