论文标题
由牛顿电位驱动的交互式方程系统的许多粒子极限
Many-particle limit for a system of interaction equations driven by Newtonian potentials
论文作者
论文摘要
我们考虑了一个由一维牛顿电位驱动的非本地相互作用的两个物种的离散粒子系统，并具有反击的自我交织和有吸引力的跨界。在有限维框架中提供了合适的存在理论之后，我们在碰撞时探索了粒子系统的行为，并使用具有粒子簇的初始数据分析解决方案的行为。随后，我们证明，与粒子系统相关的经验度量将在适当的测量意义上具有非局部相互作用项的两个部分微分方程（PDE）的系统收敛。后一个结果使用粒子轨迹对密度的分段恒定重建的$ l^m $ norms的均匀估计。
We consider a discrete particle system of two species coupled through nonlocal interactions driven by the one-dimensional Newtonian potential, with repulsive self-interaction and attractive cross-interaction. After providing a suitable existence theory in a finite-dimensional framework, we explore the behaviour of the particle system in case of collisions and analyse the behaviour of the solutions with initial data featuring particle clusters. Subsequently, we prove that the empirical measure associated to the particle system converges to the unique 2-Wasserstein gradient flow solution of a system of two partial differential equations (PDEs) with nonlocal interaction terms in a proper measure sense. The latter result uses uniform estimates of the $L^m$-norms of a piecewise constant reconstruction of the density using the particle trajectories.
