Zero-Viscosity Limit of the Linearized Compressible Navier-Stokes Equations with Highly Oscillatory Forces in the Half-Plane

We study the asymptotic expansion of solutions to the linearized compressible Navier-Stokes equations with highly oscillatory forces in the half-plane with nonslip boundary conditions for small viscosity. The wave length of oscillations is assumed to be proportional to the square root of the viscosity. By means of asymptotic analysis, we deduce that the zero-viscosity limit of solutions satisfies a linearized Euler system away from the boundary, and oscillations are propagated in a way of linear geometric optics in free space. In a small neighborhood of boundary, a boundary layer appears and satisfies a linearized Prandtl system. There is an interaction between the boundary layer and highly oscillatory waves near the boundary, which is described by an initial-boundary value problem for a Poisson-Prandtl coupled system. Finally, by using the energy method and mode analysis, we obtain the well-posedness of this Poisson-Prandtl coupled problem, and a rigorous theory on the asymptotic analysis of the zero-viscosity limit.