Explicit spectral gap estimates for the linearized Boltzmann operator modeling reactive gaseous mixtures

TL;DR

This paper derives explicit spectral gap estimates for the linearized Boltzmann operator in reactive gaseous mixtures, enabling quantification of convergence rates to equilibrium.

Contribution

It extends spectral gap estimates from elastic collisions to reactive mixtures, providing explicit bounds for the linearized chemical Boltzmann operator.

Findings

Explicit negative upper bounds for the Dirichlet form of the linearized operator.

Quantitative estimates of convergence rates to chemical equilibrium.

Application to multi-species gaseous mixture models.

Abstract

We consider hard-potential cutoff multi-species Boltzmann operators modeling microscopic binary elastic collisions and bimolecular reversible chemical reactions inside a gaseous mixture. We prove that the spectral gap estimate derived for the linearized elastic collision operator can be exploited to deduce an explicit negative upper bound for the Dirichlet form of the linearized chemical Boltzmann operator. Such estimate may be used to quantify explicitly the rate of convergence of close-to-equilibrium solutions to the reactive Boltzmann equation toward the global chemical equilibrium of the mixture.