  The rigorous justification of the hydrodynamic limits of kinetic equations in
bounded domains has been actively investigated in recent years. In spite of the
progress for the diffuse-reflection boundary case, the more challenging in-flow
boundary case, in which the leading-order boundary layer effect is
non-negligible, still remains open.
  In this work, we consider the stationary and evolutionary Boltzmann equation
with the in-flow boundary in general (convex or non-convex) bounded domains,
and demonstrate their incompressible Navier-Stokes-Fourier (INSF) limits in
$L^2$.
  Our method relies on a novel and surprising gain of
$\varepsilon^{\frac{1}{2}}$ in the kernel estimate, which is rooted from a key
cancellation of delicately chosen test functions and conservation laws. We also
introduce the boundary layer with grazing-set cutoff and investigate its BV
regularity estimates to control the source terms of the remainder equation with
the help of Hardy's inequality.
