Landau damping in finite regularity for unconfined systems with screened interactions

TL;DR

This paper proves Landau damping for the Vlasov equation with screened Coulomb interactions in unbounded space, extending results to initial data with finite regularity and highlighting a dispersive mechanism that mitigates plasma echo resonance.

Contribution

It establishes Landau damping for the collisionless Vlasov equation with $L^1$ interaction potentials in unbounded space, allowing for initial data in Sobolev, Gevrey, and analytic classes, unlike previous confined cases.

Findings

Landau damping holds for localized disturbances in unbounded space.

Finite regularity is sufficient due to dispersive effects reducing plasma echo resonance.

Results include physical screened Coulomb interactions.

Abstract

We prove Landau damping for the collisionless Vlasov equation with a class of $L^1$ interaction potentials (including the physical case of screened Coulomb interactions) on $\mathbb R^3_x \times \mathbb R^3_v$ for localized disturbances of an infinite, homogeneous background. Unlike the confined case $\mathbb T^3_x \times \mathbb R_v^3$, results are obtained for initial data in Sobolev spaces (as well as Gevrey and analytic classes). For spatial frequencies bounded away from zero, the Landau damping of the density is similar to the confined case. The finite regularity is possible due to an additional dispersive mechanism available on $\mathbb R_x^3$ which reduces the strength of the plasma echo resonance.