The Hilbert expansion of the Boltzmann equation in the incompressible Euler level in a channel

The study of the hydrodynamic limit of the Boltzmann equation with physical boundary is a challenging problem due to the appearance of the viscous and Knudsen boundary layers. In this paper, the hydrodynamic limit from the Boltzmann equation with the specular reflection boundary condition to the incompressible Euler equations in a channel is investigated. Based on the multi-scaled Hilbert expansion, the equations with boundary conditions and compatibility conditions for interior solutions, and viscous, and Knudsen boundary layers are derived under different scaling, respectively. Then, some uniform estimates for the interior solutions, and viscous, and Knudsen boundary layers are established. With the help of the L-2-L-infinity 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.