Tuesday,  February 17, 2015, 4:00 p.m.
 Chanwoo Kim (UW Madison)
 Hydrodynamic limit from the Boltzmann to the Navier-Stokes-Fourier.



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 $L^{\frac{3}{2}+}{(\Omega)}$ external force 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 its non-grazing component, which is critical to close
our program in 3D. Our results 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.