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arXiv · 1502.05324

Stationary solutions to the Boltzmann equation in the Hydrodynamic limit

Raffaele Esposito·Yan Guo·Chanwoo Kim·Rossana Marra
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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 $L^{2}-L^{\infty }$ 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 {asymptotical} stability of steady Boltzmann solutions as well as the derivation of the {unsteady} Navier-Stokes Fourier system}.

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Raffaele Esposito, Yan Guo, Chanwoo Kim, Rossana Marra. 2016-04-07. Stationary solutions to the Boltzmann equation in the Hydrodynamic limit. https://doi.org/10.1007/s40818-017-0037-5

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