Stationary solutions to the Boltzmann equation in the Hydrodynamic limit
Abstract
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 approach with new estimates for the hydrodynamic part 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}.
Keywords
Cite
@article{arxiv.1502.05324,
title = {Stationary solutions to the Boltzmann equation in the Hydrodynamic limit},
author = {Raffaele Esposito and Yan Guo and Chanwoo Kim and Rossana Marra},
journal= {arXiv preprint arXiv:1502.05324},
year = {2018}
}
Comments
New $L^6$ bound of the hydrodynamic part is added, which allows direct convergence to Navier-Stokes-Fourier system in Theorem 1.1 and Theorem 1.3