Observation estimate for kinetic transport equation by diffusion approximation

TL;DR

This paper establishes an observation estimate for kinetic transport equations using diffusion approximation, linking the small mean free path and initial data frequency to the ability to observe solutions, with implications for inverse problems.

Contribution

It introduces a novel observation estimate for the neutron transport and Fokker-Planck equations based on diffusion approximation and heat kernel analysis.

Findings

Derived an observation estimate depending on mean free path and initial data frequency.

Proposed a direct proof for observing parabolic equations at one time.

Connected kinetic transport properties with diffusion approximation techniques.

Abstract

We study the unique continuation property for the neutron transport equation and for a simplified model of the Fokker-Planck equation in a bounded domain with absorbing boundary condition. An observation estimate is derived. It depends on the smallness of the mean free path and the frequency of the velocity average of the initial data. The proof relies on the well known diffusion approximation under convenience scaling and on basic properties of this diffusion. Eventually we propose a direct proof for the observation at one time of parabolic equations. It is based on the analysis of the heat kernel.