Stationary solutions to the Boltzmann equation in the Hydrodynamic limit

Raffaele Esposito, Yan Guo, Chanwoo Kim, Rossana Marra

Published 2015-02-18Version 1

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 $L^{\frac{3}{2}+}{(\Omega)}$ external field and a small $H^{1+}{(\partial\Omega)}$ boundary temperature variation for the diffuse boundary condition. Avoiding the boundary layer correction to cope with general geometry, we employ a recent quantitative $L^{2}-L^{\infty}$ approach with crucial almost $L^{3}$ estimates for the hydrodynamic part $\mathbf{P}f$ of the distribution function. Such a gain of integrability is established via an extension lemma for the non-grazing component of the distribution function, which is critical to close our program in 3D. 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.