On the multi-species Boltzmann equation with uncertainty and its stochastic Galerkin approximation

TL;DR

This paper analyzes the multi-species Boltzmann equation with uncertain initial data and collision kernel, establishing well-posedness, exponential decay to equilibrium, and spectral gap estimates for a stochastic Galerkin approximation.

Contribution

It extends analytical tools to prove well-posedness and long-time behavior for the multi-species Boltzmann equation with uncertainty, including the first sensitivity system analysis.

Findings

Proved well-posedness of the stochastic multi-species Boltzmann equation.

Established exponential decay to equilibrium for the solution.

Derived spectral gap estimates for the stochastic Galerkin system.

Abstract

In this paper the nonlinear multi-species Boltzmann equation with random uncertainty coming from the initial data and collision kernel is studied. Well-posedness and long-time behavior - exponential decay to the global equilibrium - of the analytical solution, and spectral gap estimate for the corresponding linearized gPC-based stochastic Galerkin system are obtained, by using and extending the analytical tools provided in [M. Briant and E. S. Daus, {\it Arch. Ration. Mech. Anal.}, 3, 1367-1443, 2016] for the deterministic problem in the perturbative regime, and in [E. S. Daus, S. Jin and L. Liu, {\it Kinet. Relat. Models}, 12, 909-922, 2019] for the single-species problem with uncertainty. The well-posedness result of the sensitivity system presented here has not been obtained so far even for the single-species case.