Weak collision effect on nonlinear Landau damping for the Vlasov-Poisson-Fokker-Planck system

TL;DR

This work analyzes the Vlasov-Poisson-Fokker-Planck system on a torus near the Maxwellian under weak collisions (small diffusion ν). It develops a Gevrey-$1/s$ framework with time-dependent multipliers and a bootstrap scheme to prove global stability and enhanced dissipation, delineating a sharp regularity threshold: for $1/s\le 3$ ($s\ge 1/3$) the perturbation size can be ν-independent, while for $1/s>3$ ($s<1/3$) the perturbation must satisfy $\varepsilon\lesssim ν^{(1-3s)/(3-3s)}$. The analysis combines density-transport coupling, Volterra/Density-Resolvant control, high-regularity energy estimates, and hypocoercivity to balance phase mixing, enhanced dissipation, and plasma echoes. These results illuminate the interplay between phase mixing, diffusion, and nonlinear echoes, providing a rigorous picture of weak-collision effects on nonlinear Landau damping with potential implications for kinetic models with small collisionality.

Abstract

We investigate the impact of weak collisions on Landau damping in the Vlasov-Poisson-Fokker-Planck system on a torus, specifically focusing on its proximity to a Maxwellian distribution. In the case where the Gevrey index satisfies $\frac{1}{s}\leq3$, we establish the global stability and enhanced dissipation of small initial data, which remain unaffected by the small diffusion coefficient $ν$. For Gevrey index $\frac{1}{s}>3$, we prove the global stability and enhanced dissipation of initial data, whose size is on the order of $O(ν^\frac{1-3s}{3-3s})$. Our analysis provides insights into the effects of phase mixing, enhanced dissipation, and plasma echoes.

Theorems & Definitions (14)

• Theorem 1.1
• Remark 1.1
• Proposition 1.1
• proof
• Proposition 1.2
• proof
• Proposition 2.1
• Proposition 2.2
• proof
• proof