Stationary solutions to the Boltzmann equation in the Hydrodynamic limit

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 $L^2-L^{\mathrm{\infty }}$ approach with new $L^6$ estimates for the hydrodynamic part $\mathbf{P}f$ 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-Stokes Fourier system}.