Explicit spectral gap estimates for the linearized Boltzmann operator modeling reactive gaseous mixtures
Andrea Bondesan, Bao Quoc Tang
TL;DR
The paper proves an explicit spectral gap for the linearized Boltzmann operator of a four-species reactive gas, $\mathbf{L}=\mathbf{L}^{EL}+\mathbf{L}^{CH}$, by leveraging the known gap for the elastic part and rigorously controlling the reactive component. It splits the operator into elastic and chemical pieces, analyzes their kernels, and introduces an energy functional on the elastic kernel to quantify the cross-effects of reactions. An explicit bound for the gap $\lambda$ is derived in terms of physical parameters (masses, concentrations, reaction energy) and collision kernels, yielding a quantitative rate of convergence to the global chemical equilibrium in $L^2(\mathbb{R}^3, \pmb{\mu}^{-1/2})$. This result provides a foundational step toward quantitative nonlinear theory and hydrodynamic limits for reactive mixtures, with potential extensions to more complex kinetics and far-from-equilibrium regimes.
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.