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We establish the incompressible Navier-Stokes-Fourier limit for solutions tothe Boltzmann equation with a general cut-off collision kernel in a boundeddomain. Appropriately scaled families of DiPerna-Lions-(Mischler) renormalizedsolutions with Maxwell reflection boundary conditions are shown to havefluctuations that converge as the Knudsen number goes to zero. Every limitpoint is a weak solution to the Navier-Stokes-Fourier system with differenttypes of boundary conditions depending on the ratio between the accommodationcoefficient and the Knudsen number. The main new result of the paper is thatthis convergence is strong in the case of Dirichlet boundary condition. Indeed,we prove that the acoustic waves are damped immediately, namely they are dampedin a boundary layer in time. This damping is due to the presence of viscous andkinetic boundary layers in space. As a consequence, we also justify the firstcorrection to the infinitesimal Maxwellian that one obtains from theChapman-Enskog expansion with Navier-Stokes scaling.  This extends the work of Golse and Saint-Raymond \cite{Go-Sai04, Go-Sai05}and Levermore and Masmoudi \cite{LM} to the case of a bounded domain. The caseof a bounded domain was considered by Masmoudi and Saint-Raymond \cite{M-S} forlinear Stokes-Fourier limit and Saint-Raymond \cite{SRM} for Navier-Stokeslimit for hard potential kernels. Both \cite{M-S} and \cite{SRM} didn't studythe damping of the acoustic waves. This paper extends the result of \cite{M-S}and \cite{SRM} to the nonlinear case and includes soft potential kernels. Moreimportantly, for the Dirichlet boundary condition, this work strengthens theconvergence so as to make the boundary layer visible. This answers an openproblem proposed by Ukai \cite{Ukai}.

Boundary layers and incompressible Navier-Stokes-Fourier limit of the Boltzmann Equation in Bounded Domain (I)