Convergence to equilibrium for linear spatially homogeneous Boltzmann equation with hard and soft potentials: a semigroup approach in $L^1$-spaces

TL;DR

This paper studies the long-term behavior of solutions to the linear Boltzmann equation with hard and soft potentials using semigroup theory in weighted L^1 spaces, establishing exponential and algebraic convergence to equilibrium.

Contribution

It provides a new semigroup-based proof of exponential convergence for hard potentials and demonstrates convergence to equilibrium with algebraic rates for soft potentials in L^1 spaces.

Findings

Exponential convergence to equilibrium for hard potentials.

Convergence to equilibrium with algebraic rate for soft potentials.

Utilizes weak-compactness and ergodic projection methods in L^1 spaces.

Abstract

We investigate the large time behavior of solutions to the spatially homogeneous linear Boltzmann equation from a semigroup viewpoint. Our analysis is performed in some (weighted) $L^{1}$-spaces. We deal with both the cases of hard and soft potentials (with angular cut-off). For hard potentials, we provide a new proof of the fact that, in weighted $L^{1}$-spaces with exponential or algebraic weights, the solutions converge exponentially fast towards equilibrium. Our approach uses weak-compactness arguments combined with recent results of the second author on positive semigroups in $L^{1}$-spaces. For soft potentials, in $L^{1}$-spaces, we exploits the convergence to ergodic projection for perturbed substochastic semigroup to show that, for very general initial datum, solutions to the linear Boltzmann equation converges to equilibrium in large time. Moreover, for a large class of initial data, we also prove that the convergence rate is at least algebraic. Notice that, for soft potentials, no exponential rate of convergence is expected because of the absence of spectral gap.