$$\text {L}^2$$-Hypocoercivity and Large Time Asymptotics of the Linearized Vlasov–Poisson–Fokker–Planck System

Abstract

This paper is devoted to the linearized Vlasov-Poisson-Fokker-Planck system in presence of an external potential of confinement. We investigate the large time behaviour of the solutions using hypocoercivity methods and a notion of scalar product adapted to the presence of a Poisson coupling. Our framework provides estimates which are uniform in the diffusion limit. As an application in a simple case, we study the one-dimensional case and prove the exponential convergence of the nonlinear Vlasov-Poisson-Fokker-Planck system without any small mass assumption.

Sharpening of decay rates in Fourier based hypocoercivity methods

“…Fig 5. The curves 𝑠 ↦ → 𝜆 2 (𝑠) and 𝑠 ↦ → λ2 (𝑠) have the same asymptotic behaviour as 𝑠 → 0 + and as 𝑠 → +∞.…”

“…Here we assume that the unique steady state is 𝐹 ∞ = 0 otherwise we have to replace 𝐹 (𝑡, •) by 𝐹 (𝑡, •) − 𝐹 ∞ and 𝐹 0 by 𝐹 0 − 𝐹 ∞ in (5). The strategy of [21], later extended in [15], is to prove that for any 𝛿 > 0 small enough, we have…”

“…The choice of 𝑧/(1 + 𝑧) picks a specific scale and one may wonder if 𝑧/(𝜀 + 𝑧) would not be a better choice for some value of 𝜀 > 0 to be determined. By "better", we simply have in mind to get a larger decay rate as 𝑡 → +∞, without trying to optimize on the constant 𝐶 in (5). It turns out that the answer is negative, as 𝜀 can be scaled out.…”

“…The twist is built upon a non-local term associated with the spectral gap of the diffusion operator obtained in the diffusion limit and controls the relaxation of the macroscopic part in the limiting diffusion equation, that is, the projection of the distribution function on the orthogonal of the kernel of the dissipative part of the evolution operator. The motivating applications are kinetic transport models with diffusive macroscopic dynamics, see, e.g., [19,18,24,30,5,22], where the results yield decay estimates in an L 2 setting.…”

Functional inequalities: nonlinear flows and entropy methods as a tool for obtaining sharp and constructive results

“…In L 2 hypocoercive approaches, there is a simple strategy: if the kinetic equation admits a diffusion limit (under the appropriate parabolic scaling) whose asymptotic behaviour is governed by a functional inequality, then the corresponding rates of convergence or decay can be reimported in the kinetic equation: see for instance [25,27,26]. This is even true for systems with a non-local Poisson coupling, as shown in [1]. However, identifying sharp rates and relating nonlinear models with their linearized counterparts, as can be done in the above examples of linear and nonlinear parabolic equations, is not done yet, even in the simplest benchmark cases considered in [4].…”