Time-asymptotic behavior of the Boltzmann equation with random inputs in whole space and its stochastic Galerkin approximation

TL;DR

This paper studies the long-time behavior of the Boltzmann equation with random inputs, showing that derivatives decay polynomially over time and analyzing the stochastic Galerkin method's error decay.

Contribution

It provides the first analysis of the time-asymptotic behavior of the Boltzmann equation with randomness and its stochastic Galerkin approximation in the whole space.

Findings

Higher-order derivatives decay polynomially over time.

Stochastic Galerkin method's numerical error also decays polynomially.

Results apply to initial data and collision kernel uncertainties.

Abstract

We consider the Boltzmann equation with random uncertainties arising from the initial data and collision kernel in the {\it whole space}, along with their stochastic Galerkin (SG) approximations. By employing Green's function method, we show that, the higher-order derivatives of the solution with respect to the random variable exhibit polynomial decay over time. These results are then applied to analyze the SG method for the SG system and to demonstrate the polynomial decay of the numerical error over time.