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Starting from the local-in-time classical solution to the compressible Eulersystem with impermeable boundary condition in half-space, by employing thecoupled weak viscous layers (governed by linearized compressible Prandtlequations with Robin boundary condition) and linear kinetic boundary layers,and the analytical tools in \cite{Guo-Jang-Jiang-2010-CPAM} and some newboundary estimates both for Prandtl and Knudsen layers, we proved thelocal-in-time existence of Hilbert expansion type classical solutions to thescaled Boltzmann equation with Maxwell reflection boundary condition withaccommodation coefficient $\alpha_\varepsilon=O(\sqrt{\varepsilon})$ when theKnudsen number $\varepsilon$ small enough. As a consequence, this justifies thecorresponding case of formal analysis in Sone's books \cite{Sone-2002book,Sone-2007-Book}. This also extends the results in \cite{GHW-2020} from specularto 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.

Compressible Euler limit from Boltzmann equation with Maxwell reflection boundary condition in half-space