  We consider the Boltzmann equations with cutoff collision kernels in bounded
domains. For the initial data with finite physical bounds, we prove the
existence of global-in-time renormalized solutions in the sense of
DiPerna-Lions endowed with incoming boundary condition. Moreover, we justify
the limit as the Knudsen number $\epsilon\rightarrow 0$ to Leray solutions of
the incompressible Navier-Stokes-Fourier equations with homogeneous Dirichlet
conditions from renormalized solutions of the scaled Boltzmann equations when
the incoming data are close to the global Maxwellian in the sense of the
boundary relative entropy of order $O(\epsilon^3)$
