  The study of hydrodynamic limit of the Boltzmann equation with physical
boundary is a challenging problem due to appearance of the viscous and Knudsen
boundary layers. In this paper, the hydrodynamic limit from the Boltzmann
equation with specular reflection boundary condition to the incompressible
Euler in a channel is investigated. Based on the multiscaled Hilbert expansion,
the equations with boundary conditions and compatibility conditions for
interior solutions, viscous and Knudsen boundary layers are derived under
different scaling, respectively. Then some uniform estimates for the interior
solutions, viscous and Knudsen boundary layers are established. With the help
of $L^2-L^\infty$ framework and the uniform estimates obtained above, the
solutions to the Boltzmann equation are constructed by the truncated Hilbert
expansion with multiscales, and hence the hydrodynamic limit in the
incompressible Euler level is justified.
