Hypocoercivity and reaction-diffusion limit for a nonlinear generation-recombination model

TL;DR

This paper rigorously derives a nonlinear reaction-diffusion system from a kinetic model of a two-species gas mixture with generation and recombination, proving exponential decay to equilibrium using hypocoercivity without smallness assumptions.

Contribution

It provides the first rigorous derivation of a nonlinear reaction-diffusion system from a kinetic model and establishes hypocoercivity results for nonlinear kinetic problems without smallness constraints.

Findings

Established macroscopic reaction-diffusion limit from kinetic model

Proved exponential decay to equilibrium for the kinetic system

Achieved hypocoercivity estimates without smallness assumptions

Abstract

A reaction-kinetic model for a two-species gas mixture undergoing pair generation and recombination reactions is considered on a flat torus. For dominant scattering with a non-moving constant-temperature background the macroscopic limit to a reaction-diffusion system is carried out. Exponential decay to equilibrium is proven for the kinetic model by hypocoercivity estimates. This seems to be the first rigorous derivation of a nonlinear reaction-diffusion system from a kinetic model as well as the first hypocoercivity result for a nonlinear kinetic problem without smallness assumptions. The analysis profits from uniform bounds of the solution in terms of the equilibrium velocity distribution.