Incompressible hydrodynamic approximation with viscous heating to the Boltzmann equation

TL;DR

This paper rigorously justifies the incompressible hydrodynamic approximation of the Boltzmann equation with viscous heating, demonstrating exponential decay of the remainder under specific initial conditions.

Contribution

It provides a mathematical proof of the approximation in an $L^2\cap L^\infty$ setting, including the determination of diffusive coefficients and decay estimates.

Findings

Exponential decay of the solution remainder over time.

Validation of the incompressible Navier-Stokes-Fourier system with viscous heating as an approximation.

Explicit determination of diffusive coefficients and super Burnett functions.

Abstract

The incompressible Navier-Stokes-Fourier system with viscous heating was first derived from the Boltzmann equation in the form of the diffusive scaling by Bardos-Levermore-Ukai-Yang (2008). The purpose of this paper is to justify such an incompressible hydrodynamic approximation to the Boltzmann equation in $L^2\cap L^\infty$ setting in a periodic box. Based on an odd-even expansion of the solution with respect to the microscopic velocity, the diffusive coefficients are determined by the incompressible Navier-Stokes-Fourier system with viscous heating and the super Burnett functions. More importantly, the remainder of the expansion is proven to decay exponentially in time via an $L^2-L^\infty$ approach on the condition that the initial data satisfies the mass, momentum and energy conversation laws.