Pointwise Description for the Linearized Fokker–Planck–Boltzmann Model

In this paper, we study the pointwise (in the space variable) behavior of the linearized Fokker–Planck–Boltzmann model for nonsmooth initial perturbations. The result reveals both the fluid and kinetic aspects of this model. The fluid-like waves are constructed as the long-wave expansion in the spectrum of the Fourier modes for the space variable, and it has polynomial time decay rate. We design a Picard-type iteration for constructing the increasingly regular kinetic-like waves, which are carried by the transport equations and have exponential time decay rate. The Mixture Lemma plays an important role in constructing the kinetic-like waves, this lemma was originally introduced by Liu-Yu (Commun Pure Appl Math 57:1543–1608, 2004) for Boltzmann equation, but the Fokker–Planck term in this paper creates some technical difficulties.