Stationary solutions to the Boltzmann equation in the hydrodynamic limit

Despite its conceptual and practical importance, a rigorous derivation of 2 the steady incompressible Navier–Stokes–Fourier system from the Boltzmann theory 3 has been an outstanding open problem for general domains in 3D. We settle this 4 open question in the affirmative, in the presence of a small external field and a small 5 boundary temperature variation for the diffuse boundary condition. We employ a recent 6 quantitative L 2 – L ∞ approach with new L 6 estimates for the hydrodynamic part P f 7 of the distribution function. Our results also imply the validity of Fourier law in the 8 hydrodynamical limit, and our method leads to an asymptotical stability of steady 9 Boltzmann solutions as well as the derivation of the unsteady Navier–Stokes–Fourier 10 system.