Incompressible hydrodynamic approximation with viscous heating to the Boltzmann equation

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- . The purpose of this paper is to justify such an incompressible hydrodynamic approximation to the Boltzmann equation in L 2 ∩ L ∞ 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 ∞ approach on the condition that the initial data satisfies the mass, momentum and energy conversation laws.2010 Mathematics Subject Classification. 35Q20, 35Q79, 35C20.

Hilbert expansion of the Boltzmann equation with specular boundary condition in half-space

“…There have been extensive research efforts and literature to derive the incompressible Navier-Stokes system, see [2,3,5,8,11,12,13,14,15,18,24,32,34,36,39,46] and the references cited therein.…”

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