  The study of flows over an obstacle is one of the fundamental problems in
fluids. In this paper we establish the global validity of the diffusive limit
for the Boltzmann equations to the Navier-Stokes-Fourier system in an exterior
domain. To overcome the well-known difficulty of the lack of Poincare's
inequality in unbounded domain, we develop a new $L^2-L^6$ splitting to extend
$L^2-L^\infty$ framework into the unbounded domain.
