The study of flows over an obstacle is one of the fundamental problems in fluids. In this talk 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 the unbounded domain, we develop a new $L^2-L^6$ splitting to extend the $L^2-L^\infty$ framework into the unbounded domain.
