Stationary solutions to the Boltzmann equation in the Hydrodynamic limit

TL;DR

This paper proves the rigorous derivation of the steady incompressible Navier-Stokes-Fourier system from the Boltzmann equation in 3D domains with small external fields and boundary temperature variations, confirming the Fourier law and stability of solutions.

Contribution

It provides the first rigorous derivation of the steady Navier-Stokes-Fourier system from Boltzmann in 3D with boundary effects, using a novel $L^{2}-L^{inity}$ approach.

Findings

Validation of Fourier law in the hydrodynamic limit.

Asymptotic stability of steady Boltzmann solutions.

Derivation of unsteady Navier-Stokes Fourier system.

Abstract

Despite its conceptual and practical importance, the rigorous derivation of the steady incompressible Navier-Stokes-Fourier system from the Boltzmann theory has been {an} outstanding {open problem} for general domains in 3D. We settle this open question in {the} affirmative, in the presence of a small external field and a small boundary temperature variation for the diffuse boundary condition. We employ a recent quantitative $L^{2}-L^{\infty }$ approach with new $L^{{6}}$ estimates for the hydrodynamic part $\mathbf{P}f$ of the distribution function. Our results also imply the validity of Fourier law in the hydrodynamical limit, and our method {leads to {asymptotical} stability of steady Boltzmann solutions as well as the derivation of the {unsteady} Navier-Stokes Fourier system}.