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It is well known that the full compressible Navier-Stokes equations can bededuced via the Chapman-Enskog expansion from the Boltzmann equation as thefirst-order correction to the Euler equations with viscosity andheat-conductivity coefficients of order of the Knudsen number $\epsilon>0$. Inthe paper, we carry out the rigorous mathematical analysis of the compressibleNavier-Stokes approximation for the Boltzmann equation regarding theinitial-boundary value problems in general bounded domains. The main goal is tomeasure the uniform-in-time deviation of the Boltzmann solution with diffusivereflection boundary condition from a local Maxwellian with its fluid quantitiesgiven by the solutions to the corresponding compressible Navier-Stokesequations with consistent non-slip boundary conditions whenever $\epsilon>0$ issmall enough. Specifically, it is shown that for well chosen initial dataaround constant equilibrium states, the deviation weighted by a velocityfunction is $O(\epsilon^{1/2})$ in $L^\infty_{x,v}$ and $O(\epsilon^{3/2})$ in$L^2_{x,v}$ globally in time. The proof is based on the uniform estimates forthe remainder in different functional spaces without any spatial regularity.One key step is to obtain the global-in-time existence as well asuniform-in-$\epsilon$ estimates for regular solutions to the full compressibleNavier-Stokes equations in bounded domains when the parameter $\epsilon>0$ isinvolved in the analysis.

Compressible Navier-Stokes approximation for the Boltzmann equation in bounded domains