Asymptotic Analysis For Interaction Of Boundary Layers And Highly Oscillatory Waves In Compressible N-S Equations

The boundary layer problem plays a very important role in many applied fieldssuch as physics, mechanics and engineering. Up to now, most of works are merelyfocused on experiment analysis and numerical simulations,and there are few results inthe mathematical theories. In recent years, there have been some meaningful results onboundary layers of the incompressible Navier-Stokes equations, but there is few workon the compressible boundary layers. Obviously, from the study on the mathematicaltheory of the boundary layers, we could have a clear knowledge and good command tothe motion of ?uids, and these theoretical results could be applied in some practicalproblems.In this thesis, we study the asymptotic behavior of the interaction between bound-ary layers and highly oscillatory waves in linearized compressible Navier-Stokes equa-tions with no-slip boundary condition when the viscosity vanishes in three space vari-ables. The wave length is assumed to be proportional to the square of the viscosity. By means of multi-scale analysis we conclude that the leading profiles of solutions canbe divided into four terms: First, the no oscillatory part of the out?ow is described bythe linearized Euler equations; Second, the oscillatory wave in the out?ow propagatesalong the character field which is tangential to the boundary, and its amplitude sat-isfies a linear degenerate parabolic equation with the second order term coming fromthe viscous term in the linearized Navier-Stokes equations; Third, the no oscillatorypart in the boundary layer is described by the linearized Prandtl equation; Fourth,the oscillatory wave in the boundary layer is described by the initial-boundary valueproblem for the Poisson-Prandtl coupled system. This result shows that the zero-viscosity limit of the solution to the linearized compressible Navier-Stokes equationswith highly oscillatory forces satisfies the linearized Euler equations away from theboundary, and the oscillation is propagated in a way of linear geometric optics in freespace.The boundary layer is of Prandtl type as usual, but there is an interaction be-tween the boundary layer and the highly oscillatory waves propagated on the boundary.