Hydrodynamic limit of the Vlasov-Poisson-Fokker-Planck system in low-field regime

TL;DR

This paper rigorously derives and justifies the hydrodynamic limit of the Vlasov-Poisson-Fokker-Planck system in the low-field regime, showing convergence to the Drift-Diffusion-Poisson system using advanced energy estimates.

Contribution

It provides a rigorous mathematical proof of the hydrodynamic limit from VPFP to DDP systems, overcoming nonlinear coupling challenges with refined energy methods.

Findings

Established pointwise convergence from VPFP to DDP system.

Developed high-order energy estimates for uniform bounds.

Proved global well-posedness of solutions without a priori assumptions.

Abstract

In this paper, we study the hydrodynamic limit of the scaled Vlasov-Poisson-Fokker-Planck (VPFP) system in the low-field regime. By employing the moment method, we formally derive the corresponding Drift-Diffusion-Poisson (DDP) system. Furthermore, we rigorously justify the pointwise convergence from the VPFP system to the DDP system through delicate high-order energy estimates based on the Macro-Micro decomposition. The main difficulty lies in controlling the nonlinear coupling between the kinetic and electrostatic fields and establishing uniform bounds with respect to the scaling parameter. These challenges are overcome by developing refined high-order energy methods that yield uniform energy estimates and ensure the global well-posedness of smooth solutions, without relying on any a priori assumptions for the limiting DDP system.