Behaviour of the Fokker–Planck–Boltzmann equation near a Maxwellian

Abstract

We consider the initial value problem for the Fokker–Planck–Boltzmann equation namely, viewed as the Boltzmann equation with an additional diffusion term in velocity space to describe, for instance, the transport in thermal baths of binary elastic collisional particles. The strong solution for initial data near an absolute Maxwellian is proved to exist globally in time and tends asymptotically in the \({L^\infty_y(L_\xi^1)}\) -norm to another time dependent self-similar Maxwellian in large time. The effect of the diffusion in phase space is investigated. It produces a diffusion process in velocity space and results in a heating process on the macroscopic fluid-dynamic observable, accelerating the convergence of solutions to the equilibrium of a self-similar Maxwellian at a faster time-decay rate than the Boltzmann equation. This phenomena is also observed for homogeneous Fokker–Planck–Boltzmann equations, where the time-decay rate in the \({L_\xi^1}\) -norm to the self-similar Maxwellian is proved to be faster than exponential. Moreover, the Fokker–Planck–Boltzmann equation is shown to converge (under an appropriate scaling) strongly to the Boltzmann equation in the process of the zero diffusion limit.