Justification of Diffusion limit for the Boltzmann Equation with a non-trivial Profile

TL;DR

This paper justifies the diffusion limit of the Boltzmann equation with a non-trivial profile, demonstrating convergence to a diffusion wave and revealing higher-order flow effects due to temperature gradients.

Contribution

It provides a rigorous justification of the diffusion limit for the Boltzmann equation with a non-trivial background profile under slab symmetry.

Findings

Solution converges to a diffusion wave with decay rates in Knudsen number and time.

Reveals higher-order flow effects driven by temperature gradients.

Establishes the diffusion phenomena in temperature and density.

Abstract

Under the diffusion scaling and a scaling assumption on the microscopic component, a non-classical fluid dynamic system was derived in \cite{BGLY} that is related to the system of ghost effect derived in \cite{Sone-2} in different settings. This paper aims to justify this limit system for a non-trivial background profile with slab symmetry. The result reveals not only the diffusion phenomena in the temperature and density, but also the flow of higher order in Knudsen number due to the gradient of the temperature. Precisely, we show that the solution to the Boltzmann equation converges to a diffusion wave with decay rates in both Knudsen number and time.