Stationary non equilibrium states in kinetic theory

Stationary non equilibrium solutions to the Boltzmann equation, despite their relevance in applications, are much less studied than time dependent solutions, and no general existence theory is yet available, due to serious technical difficulties. Here we review some results on the construction of stationary non equilibrium solutions, in a general domain in contact with a slightly non-homogeneous thermal reservoir, both for finite and small Knudsen number. We will describe different approaches and different techniques developed. The main focus will be on stationary solutions close to hydrodynamics. In particular, we will give an answer to the longstanding open problem of the rigorous derivation of the steady incompressible Navier-Stokes-Fourier system from the Boltzmann theory, in the presence of a small external force and diffuse boundary condition with small boundary temperature variations