Determining the collision kernel in the Boltzmann equation near the equilibrium

TL;DR

This paper addresses an inverse problem for the nonlinear Boltzmann equation near equilibrium, aiming to determine the collision kernel from boundary measurements using linearization and transform injectivity.

Contribution

It introduces a method to recover the collision kernel in the Boltzmann equation from boundary data, leveraging linearization and the Gauss-Weierstrass transform.

Findings

Successfully determines the collision kernel from the Albedo operator.

Establishes the injectivity of the Gauss-Weierstrass transform in this context.

Provides a new approach for inverse problems in kinetic theory.

Abstract

We consider an inverse problem for the nonlinear Boltzmann equation near the equilibrium. Our goal is to determine the collision kernel in the Boltzmann equation from the knowledge of the Albedo operator. Our approach relies on a linearization technique as well as the injectivity of the Gauss-Weierstrass transform.