Passage from the Boltzmann equation with Diffuse Boundary to the Incompressible Euler equation with Heat Convection

TL;DR

This paper derives the incompressible Euler equations with heat convection from the Boltzmann equation with diffuse boundary conditions in the hydrodynamic limit, using a Hilbert expansion and Green's function estimates.

Contribution

It introduces a new derivation of heat-convective Euler equations from kinetic theory, incorporating heat transfer in the hydrodynamic limit with diffuse boundary conditions.

Findings

Established a Hilbert-type expansion around Maxwellian for heat transfer

Developed a new estimate for heat flux in Navier-Stokes-Fourier system

Connected Boltzmann equation with Euler equations including heat convection

Abstract

We derive the incompressible Euler equations with heat convection with the no-penetration boundary condition from the Boltzmann equation with the diffuse boundary in the hydrodynamic limit for the scale of large Reynold number. Inspired by the recent framework in [30], we consider the Navier-Stokes-Fourier system with no-slip boundary conditions as an intermediary approximation and develop a Hilbert-type expansion of the Boltzmann equation around the global Maxwellian that allows the nontrivial heat transfer by convection in the limit. To justify our expansion and the limit, a new direct estimate of the heat flux and its derivatives in the Navier-Stokes-Fourier system is established adopting a recent Green's function approach in the study of the inviscid limit.