非正则点涡的平均场收敛性
Mean-Field Convergence of Point Vortices without Regularity
摘要：
我们考虑经典的点涡模型中的平均场标度制度，其中的速度场所经历的一个单一的点涡是成比例的平均值所产生的其余点涡的速度场。我们表明，如果在某个时候，相关的经验措施序列收敛在一个适当的意义上的概率测量密度$ω^0\in L^\infty(\mathbb{R}^2)$和具有有限的能量，作为点涡数$N\rightarrow\infty$，然后序列收敛在弱 * 拓扑的措施，以二维不可压缩欧拉方程的唯一解$ω$与初始数据$ω^0$，局部一致的时间。与以前的结果相比，我们的定理不需要对极限涡量$ω$的正则性假设，是在二维欧拉方程守恒律的水平，并提供了一个定量的收敛速度。我们的证明是基于Serfaty的调制能量方法和一个新的软化参数的组合。我们认为，我们的结果是一个平均场收敛模拟著名的定理Yudovich的整体适定性的二维欧拉在$L^1(\mathbb{R}^2)\cap L^\infty(\mathbb{R}^2)$。
英文摘要：
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 $L^1(\mathbb{R}^2)\cap L^\infty(\mathbb{R}^2)$.
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