In the study of the heat transfer in the Boltzmann theory, the basic problem is to construct solutions to the following steady problem:

Abstract.In this paper we analyze a system of N identical quantum particles in a weakcoupling regime. The time evolution of the Wigner transform of the one-particle reduced density matrix is represented by means of a perturbative series. The expansion is obtained upon iterating the Duhamel formula. For short times, we rigorously prove that a subseries of the latter, converges to the solution of the Boltzmann equation which is physically relevant in the context. In particular, we recover the transition rate as it is predicted by Fermi's Golden Rule. However, we are not able to prove that the quantity neglected while retaining a subseries of the complete original perturbative expansion, indeed vanishes in the limit: we only give plausibility arguments in this direction. The present study holds in any space dimension d ≥ 2.

We consider the time evolution of a one dimensional n-gradient continuum.\ud Our aim is to construct and analyze discrete approximations in terms of physically\ud realizable mechanical systems, referred to as microscopic because they are living\ud on a smaller space scale. We validate our construction by proving a convergence\ud theorem of the microscopic system to the given continuum, as the scale parameter\ud goes to zero

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 ∞ approach with new L 6 estimates for the hydrodynamic part Pf 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-StokesFourier system.

We study the stationary solution of the Boltzmann equation in a slab with a constant external force parallel to the boundary and complete accommodation condition on the walls at a specified temperature. We prpve that when the force is sufficiently small there exists a solution which converges, in the hydrodynamic limit, to a local Maxwellian with parameters given by the stationary solution of the corresponding compressible Navier-Stokes equations with no-slip boundary conditions. Corrections to this Maxwellian are obtained in powers of the Knudsen number with a controlled remainder.