The Compressible boundary layer and the Boltzmann equation

We consider the flow of ideal gas in half space described by the system of compressible Navier-Stokes equations. We apply the Prandtl scaling and we obtain the system of compressible Prandtl equations. In this article, a modification of the classical Chapman-Enskog method is proposed, which allows us to derive the system of compressible Prandtl equations directly from the Boltzmann equation without the use of the Knudsen-layer correction. Different types of boundary conditions are discussed.