Incompressible hydrodynamic approximation with viscous heating to the   Boltzmann equation
  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. 
Yan Guo and Shuangqian Liu
math.AP