Diffusive limit of the Vlasov-Poisson-Fokker-Planck model: quantitative and strong convergence results

TL;DR

This paper establishes quantitative and strong convergence results for the diffusive limit of the Vlasov-Poisson-Fokker-Planck model, providing explicit rates and conditions for convergence in a high-dimensional setting.

Contribution

It derives a priori estimates valid in any phase-space dimension and proves an explicit convergence rate near the optimal, extending previous results with detailed analysis of phase-space coordinates.

Findings

Strong convergence in L2 space for the diffusive limit.

Explicit convergence rate arbitrarily close to the optimal.

Results hold on unbounded time intervals with explicit growth bounds.

Abstract

This work tackles the diffusive limit for the Vlasov-Poisson-Fokker-Planck model. We derive a priori estimates which hold without restriction on the phase-space dimension and propose a strong convergence result in a L2 space. Furthermore, we strengthen previous results by obtaining an explicit convergence rate arbitrarily close to the (formal) optimal rate, provided that the initial data lies in some Lp space with p large enough. Our result holds on bounded time intervals whose size grow to infinity in the asymptotic limit with explicit lower bound. The analysis relies on identifying the right set of phase-space coordinates to study the regime of interest. In this set of coordinates the limiting model arises explicitly.