Uniform regularity for linear kinetic equations with random input based on hypocoercivity

TL;DR

This paper establishes uniform regularity results for linear kinetic equations with random inputs, demonstrating solution analyticity and convergence of numerical methods across various models and regimes using hypocoercivity.

Contribution

It introduces a general framework proving uniform regularity and solution analyticity for kinetic equations with randomness, independent of specific collision terms or distributions.

Findings

Solutions are analytic in the random variables.

Uniform regularity holds across kinetic, parabolic, and high field regimes.

Hypocoercivity is effectively used for convergence analysis.

Abstract

In this paper we study the effect of randomness in kinetic equations that preserve mass. Our focus is in proving the analyticity of the solution with respect to the randomness, which naturally leads to the convergence of numerical methods. The analysis is carried out in a general setting, with the regularity result not depending on the specific form of the collision term, the probability distribution of the random variables, or the regime the system is in, and thereby termed "uniform". Applications include the linear Boltzmann equation, BGK model, Carlemann model, among many others; and the results hold true in kinetic, parabolic and high field regimes. The proof relies on the explicit expression of the high order derivatives of the solution in the random space, and the convergence in time is mainly based on hypocoercivity, which, despite the popularity in PDE analysis of kinetic theory, has rarely been used for numerical algorithms.