Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation
Frederic Herau
TL;DR
This work analyzes an inhomogeneous linear Boltzmann equation with a confining potential and a simple relaxation collision operator Q(f)=γ(ρ μ∞−f), proving exponential convergence to the global Maxwellian in a weighted space. Using a hypocoercivity framework, the authors adopt a Hilbert-space formulation via a conjugation by $M^{-1/2}$ and build a transport-collision operator $K$, enabling spectral analysis without diffusion. They establish the existence of constants ε>0 and A>0 (explicitly bounded in terms of the potential $V$ and γ) such that a modified norm satisfies Re$(Ku,(Id+ε(L+L^*))u) \\ge (α^2)/A \,||u||^2$ for all $u$ orthogonal to $M^{1/2}$, yielding exponential decay in $L^2$ and hence in the $B^2$ space with rate $e^{−α^2 t/A}$. An entropy-decay bound $H(f|f∞)(t) \\le C e^{−α^2 t/A}$ is also derived, with discussion of cases where $e^{−V} otin L^1$ and connections to Fokker–Planck and linear inhomogeneous Boltzmann models, highlighting the broad implications of the hypocoercivity approach for inhomogeneous kinetic equations without diffusion.
Abstract
We consider an inhomogeneous linear Boltzmann equation, with an external confining potential. The collision operator is a simple relaxation toward a local Maxwellian, therefore without diffusion. We prove the exponential time decay toward the global Maxwellian, with an explicit rate of decay. The methods are based on hypoelliptic methods transposed here to get spectral information. They were inspired by former works on the Fokker-Planck equation and the main feature of this work is that they are relevant although the equation itself has no regularizing properties.