Stable determination of time-dependent collision kernel in the nonlinear Boltzmann equation

TL;DR

This paper proves a stable method for determining a time-dependent collision kernel in the nonlinear Boltzmann equation using measurements, with implications for uniqueness and inverse problem analysis.

Contribution

It introduces a logarithmic stability estimate for the inverse problem of identifying the collision kernel, employing second-order linearization and light-ray transform techniques.

Findings

Established a logarithm-type stability result.

Proved uniqueness of the collision kernel under certain conditions.

Applied advanced linearization and transform methods to inverse problems.

Abstract

We consider an inverse problem for the nonlinear Boltzmann equation with a time-dependent kernel in dimensions $n\ge 2$. We establish a logarithm-type stability result for the collision kernel from measurements under certain additional conditions. A uniqueness result is derived as an immediate consequence of the stability result. Our approach relies on second-order linearization, multivariate finite differences, as well as the stability of the light-ray transform.