Landau Damping in Finite Regularity for Unconfined Systems with Screened Interactions

Abstract

We prove Landau damping for the collisionless Vlasov equation with a class of L1 interaction potentials (including the physical case of screened Coulomb interactions) on ℝx3×ℝv3 for localized disturbances of an infinite, homogeneous background. Unlike the confined case double-struckTx3×ℝv3, results are obtained for initial data in Sobolev spaces (as well as Gevrey and analytic classes). For spatial frequencies bounded away from 0, 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 ℝx3 that reduces the strength of the plasma echo resonance.© 2017 Wiley Periodicals, Inc.

Suppression of Plasma Echoes and Landau Damping in Sobolev Spaces by Weak Collisions in a Vlasov-Fokker-Planck Equation

“…As has been done in several previous works on Landau damping and mixing in fluid mechanics [12,10,13,58,11,7,8,9], we apply a bootstrap argument. Bootstrap arguments typically require a local well-posedness and propagation of regularity result.…”

“…The above scheme is significantly more detailed than the schemes used in e.g. [12,58,11]. This is partly due to the drift term ∇ v · (vF ) in the linear Fokker-Planck operator, but is more due to the complications from the nonlinear, x-dependent coefficients in the Fokker-Planck collision operator in (1.1).…”

“…The Volterra equation (3.5) is then studied using techniques similar to [40,12,11] and the references. The results are similar to results of [54], however, here we gain the collisional decay factor e −δν 1/3 t .…”

“…[40,12]), and hence we omit the details for brevity. In order to get polynomial decay estimates, take (|k| ∂ ω ) α derivatives of (3.6) as in [11], giving…”

“…See [11] for more details in a similar setting and [40,12] for discussions on how to make the a priori estimate rigorous.…”