AI 中文总结
研究\(N\)个相互作用粒子在平均场状态下演化的一阶常微分方程系统，通过引入新方法，利用热核软化研究度量演化，证明离散系统到平均场方程解的定量收敛在多种情况下成立，且部分结果最优。
AI 中文摘要
我们考虑一类用于描述\(N\)个相互作用粒子（在欧几里得空间\(\mathbb{R}^d\)中）在平均场状态下演化的一阶常微分方程系统。所处理的相互作用类型包括幂次高达\(d + 1\)的逆幂型奇异相互作用，有吸引或排斥的，且不一定源于势——这与例如调制能量方法不同。我们引入一种新方法来证明离散系统到平均场方程解的定量收敛。该方法依赖于通过热核软化研究一种度量的演化，该度量对经验测度与其极限之间的差异进行多尺度控制。我们证明了在以下情况下所需的收敛成立：(i) 如果奇点在任何维度下是亚库仑的，或者在一维和二维中是库仑的（为此我们引入了常微分方程系统的弱解概念），则在极限方程光滑解的最大存在时间之前；(ii) 在三维及以上维度存在库仑奇点的情况下在短时间内成立；(iii) 在所有维度中超库仑相互作用下直到一个与\(N\)相关的短时间尺度成立。后两个结果被证明是最优的，因为我们证明了一类吸引相互作用在同一时间尺度内会发生碰撞。
英文摘要
We consider a general class of first order ODE systems for the evolution of $N$ interacting particles (in Euclidean space $\mathbb{R}^d$) in a mean-field regime. The class of interactions treated includes singular interactions of inverse power type up to power $d+1$, attractive or repulsive, and not necessarily deriving from a potential -- unlike, for instance, the modulated energy method. We introduce a new method to prove quantitative convergence of the discrete system to solutions of the mean-field equation. It relies on studying the evolution of a metric encoding a multiscale control of the difference between the empirical measure and its limit, via mollification by heat kernels. We prove that the desired convergence holds (i) up to the maximal time of existence of the smooth solution to the limiting equation if the singularity is sub-coulombic in any dimension, or coulombic in dimensions 1 and 2 (where, to do so, we introduce a notion of weak solution to the ODE system), or (ii) for short time in the case of Coulomb singularity in dimension 3 and above and (iii) up to a short $N$-dependent timescale for super-coulombic interactions in all dimensions. The latter two results are demonstrated to be optimal as we prove that collisions occur within the same timescale for a class of attractive interactions.
Comments105 pages
