Explicit structure of the Fokker-Planck equation with potential

Abstract

We study the pointwise (in the space and time variables) behavior of the Fokker-Planck Equation with potential. An explicit description of the solution is given, including the large time behavior, initial layer and spatially asymptotic behavior. Moreover, it is shown that the structure of the solution sensitively depends on the potential function.2010 Mathematics Subject Classification. 35Q84; 82C40.

Green's Function and Pointwise Behaviors of the Vlasov-Poisson-Fokker-Planck System

“…We now outline the main idea and make some comments on the proof of the above theorems. The results in Theorem 1.1 on the pointwise behavior of the Green's function to the VPFP system is proved based on the spectral analysis [13] and the ideas inspired by [14,15]. Indeed, we first decompose the Green function G into the lower frequency part G L and the high frequency part G H .…”

“…Since the Fourier transform of G H is not L 1 integrable in frequency space, G H can be decomposed into the singular part and the remaining smooth part. We apply the refined Picard's iteration as Fokker-Planck equation [15] to construct the singular kinetic waves J k of G H where J k (t, x) are the solutions to the Fokker-Planck equations with damping given by (1.26)-(1.27). In particular, J k (t, x)g 0 with g 0 ∈ L 2 (R 3 v ) are smooth for all (t, x, v) when t > 0, and Ĵk (t, ξ) satisfy (refer to Lemma 3.6)…”

“…Then, we introduce the Picard's iteration as VPB system [14] to construct the approximate sequences for the low frequency part, and estimate the remaining part by the energy method in frequency space. Finally, we refine the Picard's iteration in Fokker-Planck equation [15] to construct the approximate sequences (singular kinetic waves) for the high frequency part which are smooth for (t, x, v) when t > 0, and establish the pointwise behavior of the remaining part by the energy method in frequency space.…”

“…The Green's function and pointwise spacetime behavior of the Boltzmann equation were first studied in the pioneering works [16,17,18,19]. For the linear Fokker-Planck equation, the structure of the Green's function was investigated in [12,15,25,26]. Recently, Li, Yang and Zhong established the Green's function and pointwise space-time behavior of the unipolar Vlasov-Poisson-Boltzmann system in [14].…”

Finite Speed of Propagation of the Relativistic Landau and Boltzmann Equations

“…Finally, we want to emphasis that this theorem reflects the difference in essence between the relativistic kinetic equations and the classical kinetic equations. For the classical kinetic equations (Boltzmann, Landau or Fokker-Planck), note that the speed of the transport part is v, which reflects the infinite speed of propagation of the solution (see for instance the Fokker-Planck equation with potentials [16], classical Boltzmann equation with hard sphere [19], for the linear problems regarding the classical Boltzmann equation with hard potentials and soft potentials [12,15,18], and for the Vlasov-Poisson-Boltzmann system [14]). Precisely, if imposing the initial data compactly supported in the x variable, then we only have spatial decay (exponentially, subexponentially or algebraically) of the solution to the classical kinetic equations.…”