Stationary Solutions to the Boltzmann Equation in the Hydrodynamic Limit

Despite its conceptual and practical importance, a 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 \(L^{2}\)–\(L^{\infty }\) 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 an asymptotical stability of steady Boltzmann solutions as well as the derivation of the unsteady Navier–Stokes–Fourier system.