  Consider 3D Boltzmann equation in convex domains with diffusive-reflection
boundary condition. We study the hydrodynamic limits as the Knudsen number and
Strouhal number $\epsilon\rightarrow 0^+$. Using the Hilbert expansion, we
rigorously justify that the solution of stationary/evolutionary problem
converges to that of the steady/unsteady Navier-Stokes-Fourier system.
  This is the first paper to justify the hydrodynamic limits of nonlinear
Boltzmann equations with hard-sphere collision kernel in bounded domain in the
$L^{\infty}$ sense. The proof relies on a novel analysis on the boundary layer
effect with geometric correction.
  The difficulty mainly comes from three sources: 3D domain, boundary layer
regularity, and time dependence. To fully solve this problem, we introduce
several techniques: (1) boundary layer with geometric correction; (2) remainder
estimates with $L^2-L^{6}-L^{\infty}$ framework. Keywords: boundary layer;
Milne problem; geometric correction; remainder estimates.

Analysis of PDEs