Large friction-high force fields limit for the nonlinear Vlasov--Poisson--Fokker--Planck system

TL;DR

This paper analyzes the asymptotic behavior of the nonlinear Vlasov--Poisson--Fokker--Planck system under large friction and force fields, showing convergence to a nonlocal diffusive model.

Contribution

It provides a rigorous quantitative analysis demonstrating the strong convergence of solutions from the kinetic system to a diffusive aggregation-diffusion model in the large friction and force-field limit.

Findings

Weak solutions converge strongly to solutions of the diffusive model.

The analysis uses modulated macroscopic kinetic energy estimates.

The proof employs weak-strong uniqueness and Poisson equation analysis.

Abstract

We provide a quantitative asymptotic analysis for the nonlinear Vlasov--Poisson--Fokker--Planck system with a large linear friction force and high force-fields. The limiting system is a diffusive model with nonlocal velocity fields often referred to as aggregation-diffusion equations. We show that a weak solution to the Vlasov--Poisson--Fokker--Planck system strongly converges to a strong solution to the diffusive model. Our proof relies on the modulated macroscopic kinetic energy estimate based on the weak-strong uniqueness principle together with a careful analysis of the Poisson equation.