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We consider the Boltzmann equations with cutoff collision kernels in boundeddomains. For the initial data with finite physical bounds, we prove theexistence of global-in-time renormalized solutions in the sense ofDiPerna-Lions endowed with incoming boundary condition. Moreover, we justifythe limit as the Knudsen number $\epsilon\rightarrow 0$ to Leray solutions ofthe incompressible Navier-Stokes-Fourier equations with homogeneous Dirichletconditions from renormalized solutions of the scaled Boltzmann equations whenthe incoming data are close to the global Maxwellian in the sense of theboundary relative entropy of order $O(\epsilon^3)$

The Boltzmann equation with incoming boundary condition: global solutions and Navier-Stokes limit