Mean field limit for the one dimensional Vlasov-Poisson equation
Mean field limit for the one dimensional Vlasov-Poisson equation
复制标题
一维 Vlasov-Poisson 方程的平均场极限
DOI：
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发表时间：
2013
期刊：
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通讯作者：
M. Hauray
中科院分区：
文献类型：
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作者：
M. Hauray
我们考虑由库仑对或引力相互作用驱动的一维$N$粒子系统。当粒子的数量在所谓的平均场缩放中趋于无穷大时，我们正式期望收敛于Vlasov-Poisson方程。实际上，Trocheris在\cite{Tro86}中给出了收敛性的严格证明。在这里，我们将给出这个结果的一个更简单的证明，并解释为什么它隐含着所谓的“分子混沌的传播”。更确切地说，这两个结果都是一维Vlasov-Poisson方程的弱-强稳定性结果的直接结果，这个方程本身就很有趣。我们还证明了$N$粒子动力学的全局解的存在性，以及Vlasov-Poisson方程的全局解的存在性。
We consider systems of $N$ particles in dimension one, driven by pair Coulombian or gravitational interactions. When the number of particles goes to infinity in the so called mean field scaling, we formally expect convergence towards the Vlasov-Poisson equation. Actually a rigorous proof of that convergence was given by Trocheris in \cite{Tro86}. Here we shall give a simpler proof of this result, and explain why it implies the so-called "Propagation of molecular chaos". More precisely, both results will be a direct consequence of a weak-strong stability result on the one dimensional Vlasov-Poisson equation that is interesting by it own. We also prove the existence of global solutions to the $N$ particles dynamic starting from any initial positions and velocities, and the existence of global solutions to the Vlasov-Poisson equation starting from any measures with bounded first moment in velocity.