论文标题
具有较高维度的库仑相互作用的颗粒系统的平均场融合无规律
Mean-Field Convergence of Systems of Particles with Coulomb Interactions in Higher Dimensions without Regularity
论文作者
论文摘要
我们考虑在平均场比例机制中具有二进制库仑相互作用的颗粒的一阶保守系统，$ d \ geq 3 $。我们表明，如果在某个时候，相关的经验度量顺序以适当的含义收敛到概率度量$ω^0 $作为粒子数量$ n \ rightarrow \ infty $，则在短时间内，该序列在弱 - *拓扑范围内收敛，以实现最初的field pde的唯一解决方案，以最初的firt firt firt firtial datum $ω$ω^0 $ω^0。该结果扩展了我们以前的工作ARXIV：2004.04140点涡流（即$ d = 2 $）。与先前的工作ARXIV：1803.08345相反，我们的定理仅需要限制度量属于缩放量的范围函数空间，以实现平均场PDE的适当性，特别是不需要规律性。我们的证明是基于作者在Arxiv：2004.04140中首先引入的《奴隶制能源方法》和新颖的动感参数的组合。
We consider first-order conservative systems of particles with binary Coulomb interactions in the mean-field scaling regime in dimensions $d\geq 3$. We show that if at some time, the associated sequence of empirical measures converges in a suitable sense to a probability measure with bounded density $ω^0$ as the number of particles $N\rightarrow\infty$, then the sequence converges for short times in the weak-* topology for measures to the unique solution of the mean-field PDE with initial datum $ω^0$. This result extends our previous work arXiv:2004.04140 for point vortices (i.e. $d=2$). In contrast to the previous work arXiv:1803.08345, our theorem only requires the limiting measure belong to a scaling-critical function space for the well-posedness of the mean-field PDE, in particular requiring no regularity. Our proof is based on a combination of the modulated-energy method of Serfaty and a novel mollification argument first introduced by the author in arXiv:2004.04140.
