Global Existence and Decay of Solutions to the Fokker-Planck-Boltzmann Equation

TL;DR

This paper proves the global existence and decay rates of solutions to the Fokker-Planck-Boltzmann equation for small initial perturbations, using energy methods and operator coercivity.

Contribution

It establishes the first rigorous results on global solutions and decay rates for the Fokker-Planck-Boltzmann equation under Grad's cutoff assumption.

Findings

Global existence of classical solutions for small initial data

Optimal decay rates of solutions over time

Use of coercivity and energy methods in analysis

Abstract

The Cauchy problem to the Fokker-Planck-Boltzmann equation under Grad's angular cut-off assumption is investigated. When the initial data is a small perturbation of an equilibrium state, global existence and optimal temporal decay estimates of classical solutions are established. Our analysis is based on the coercivity of the Fokker-Planck operator and an elementary weighted energy method.