  Starting from the local-in-time classical solution to the compressible Euler
system with impermeable boundary condition in half-space, by employing the
coupled weak viscous layers (governed by linearized compressible Prandtl
equations with Robin boundary condition) and linear kinetic boundary layers,
and the analytical tools in \cite{Guo-Jang-Jiang-2010-CPAM} and some new
boundary estimates both for Prandtl and Knudsen layers, we proved the
local-in-time existence of Hilbert expansion type classical solutions to the
scaled Boltzmann equation with Maxwell reflection boundary condition with
accommodation coefficient $\alpha_\varepsilon=O(\sqrt{\varepsilon})$ when the
Knudsen number $\varepsilon$ small enough. As a consequence, this justifies the
corresponding case of formal analysis in Sone's books \cite{Sone-2002book,
Sone-2007-Book}. This also extends the results in \cite{GHW-2020} from specular
to Maxwell reflection boundary condition. Both of this paper and
\cite{GHW-2020} can be viewed as generalizations of Caflisch's classic work
\cite{Caflish-1980-CPAM} to the cases with boundary.
