The Vlasov–Poisson–Landau system in the weakly collisional regime

Abstract

Consider the Vlasov–Poisson–Landau system with Coulomb potential in the weakly collisional regime on a 3 3 -torus, i.e. ∂ t F ( t , x , v ) + v i ∂ x i F ( t , x , v ) + E i ( t , x ) ∂ v i F ( t , x , v ) = ν Q ( F , F ) ( t , x , v ) , E ( t , x ) = ∇ Δ − 1 ( ∫ R 3 F ( t , x , v ) d v − ∫ − T 3 ∫ R 3 F ( t , x , v ) d v d x ) , \begin{align*} \partial _t F(t,x,v) + v_i \partial _{x_i} F(t,x,v) + E_i(t,x) \partial _{v_i} F(t,x,v) = \nu Q(F,F)(t,x,v),\\ E(t,x) = \nabla \Delta ^{-1} (\int _{\mathbb R^3} F(t,x,v)\, \mathrm {d} v - {{\int }\llap {-}}_{\mathbb T^3} \int _{\mathbb R^3} F(t,x,v)\, \mathrm {d} v \, \mathrm {d} x), \end{align*} with ν ≪ 1 \nu \ll 1 . We prove that for ϵ > 0 \epsilon >0 sufficiently small (but independent of ν \nu ), initial data which are O ( ϵ ν 1 / 3 ) O(\epsilon \nu ^{1/3}) -Sobolev space perturbations from the global Maxwellians lead to global-in-time solutions which converge to the global Maxwellians as t → ∞ t\to \infty . The solutions exhibit uniform-in- ν \nu Landau damping and enhanced dissipation. Our main result is analogous to an earlier result of Bedrossian for the Vlasov–Poisson–Fokker–Planck equation with the same threshold. However, unlike in the Fokker–Planck case, the linear operator cannot be inverted explicitly due to the complexity of the Landau collision operator. For this reason, we develop an energy-based framework, which combines Guo’s weighted energy method with the hypocoercive energy method and the commuting vector field method. The proof also relies on pointwise resolvent estimates for the linearized density equation.

Taylor dispersion and phase mixing in the non-cutoff Boltzmann equation on the whole space

“…Notice that for s = 1 we have p γ,1 = 2/(2+|γ|). Thus, for the Landau collision operator we get a power p −3,1 = 2/5 which is an improvement over the 1/3 power given in [21]. We believe this improvement is related to the velocity weights we use in our energy for the hypocoercive scheme.…”

“…In [11] the enhanced dissipation effect was proved in the Vlasov-Poisson-Fokker-Planck equations on T d near a global Maxwellian, along with uniform-in-ν Landau damping estimates on the density, provided the initial velocity-weighted H σ norm of the initial condition was ν 1/3 (see earlier work by Tristani [64] that proves uniform Landau damping for the linearized problem). More recently, the case of Vlasov-Poisson-Landau equation on T d was solved by Chaturvedi, Luk, and Nguyen [21]. Due to the nonlinear echo phenomenon, the O(ν 1/3 ) threshold seems potentially sharp in Sobolev spaces [12].…”

“…Hence, the specific aim of this paper is to obtain the quantitative estimates in the limit ν → 0. Neither the methods of [21] nor [59] apply here although ideas from both are utilized; see Section 2 for an outline of the proof. Note in particular that the Landau damping is not quite the same as (1.9) for t ≫ ν −1 .…”

“…The proof on T d is significantly easier as one can adapt ideas from [21] to the Boltzmann collision operators (see especially Section 4 for what is required for this adaptation). We state the result for completeness but will not discuss the proof further.…”

“…We believe this improvement is related to the velocity weights we use in our energy for the hypocoercive scheme. In particular, we use a weight v −3/2 for terms involving v-derivatives whereas in [21] a weight v −4 is used.…”