Hilbert expansion of Boltzmann equation with soft potentials and specular boundary condition in half-space

Boundary effects play an important role in the study of hydrodynamic limits in the Boltzmann theory. We justify rigorously the validity of the hydrodynamic limit from the Boltzmann equation of soft potentials to the compressible Euler equations by the Hilbert expansion with multi-scales. Specifically, the Boltzmann solutions are expanded into three parts: interior part, viscous boundary layer and Knudsen boundary layer. Due to the weak effect of collision frequency of soft potentials, new difficulty arises when tackling the existence of Knudsen layer solutions with space decay rate, which has been overcome under some constraint conditions and losing velocity weight arguments.