In this note, we review the recent work [23] on the boundary layer and incompressible Navier-Stokes-Fourier limit of the Boltzmann equation with a general cut-off collision in a bounded domain. Appropriately scaled families of DiPerna-Lions-(Mischler) renormalized solutions with Maxwell reflection boundary conditions are shown to have fluctuations that converge as the Knudsen number goes to zero. Every limit point is a weak solution to the Navier-Stokes-Fourier system with different types of boundary conditions depending on the ratio between the accommodation coefficient and the Knudsen number.
The main new result is that this convergence is strong in the case of Dirichlet boundary condition. Indeed, we prove that the acoustic waves are damped immediately, namely they are damped in a boundary layer in time. This damping is due to the presence of viscous and kinetic boundary layers in space. As a consequence, we also justify the first correction to the infinitesimal Maxwellian that one obtains from the Chapman-Enskog expansion with Navier-Stokes scaling.
@article{SLSEDP_2015-2016____A2_0, author = {Ning Jiang and Nader Masmoudi}, title = {Boundary layers and incompressible {Navier-Stokes-Fourier} limit of the {Boltzmann} equation in a bounded domain}, journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications}, note = {talk:2}, pages = {1--16}, publisher = {Institut des hautes des scientifiques & Centre de mathtiques Laurent Schwartz, ole polytechnique}, year = {2015-2016}, doi = {10.5802/slsedp.95}, language = {en}, url = {https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.95/} }
TY - JOUR AU - Ning Jiang AU - Nader Masmoudi TI - Boundary layers and incompressible Navier-Stokes-Fourier limit of the Boltzmann equation in a bounded domain JO - Séminaire Laurent Schwartz — EDP et applications N1 - talk:2 PY - 2015-2016 SP - 1 EP - 16 PB - Institut des hautes des scientifiques & Centre de mathtiques Laurent Schwartz, ole polytechnique UR - https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.95/ DO - 10.5802/slsedp.95 LA - en ID - SLSEDP_2015-2016____A2_0 ER -
%0 Journal Article %A Ning Jiang %A Nader Masmoudi %T Boundary layers and incompressible Navier-Stokes-Fourier limit of the Boltzmann equation in a bounded domain %J Séminaire Laurent Schwartz — EDP et applications %Z talk:2 %D 2015-2016 %P 1-16 %I Institut des hautes des scientifiques & Centre de mathtiques Laurent Schwartz, ole polytechnique %U https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.95/ %R 10.5802/slsedp.95 %G en %F SLSEDP_2015-2016____A2_0
Ning Jiang; Nader Masmoudi. Boundary layers and incompressible Navier-Stokes-Fourier limit of the Boltzmann equation in a bounded domain. Séminaire Laurent Schwartz — EDP et applications (2015-2016), Talk no. 2, 16 p. doi : 10.5802/slsedp.95. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.95/
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