  A semi-explicit formula of solution to the boundary layer system for thermal
layer derived from the compressible Navier-Stokes equations with the non-slip
boundary condition when the viscosity coefficients vanish is given, in
particular in three space dimension. In contrast to the inviscid Prandtl system
studied by [7] in two space dimension, the main difficulty comes from the
coupling of the velocity field and the temperature field through a degenerate
parabolic equation. The convergence of these boundary layer equations to the
inviscid Prandtl system is justified when the initial temperature goes to a
constant. Moreover, the time asymptotic stability of the linearized system
around a shear flow is given, and in particular, it shows that in three space
dimension, the asymptotic stability depends on whether the direction of
tangential velocity field of the shear flow is invariant in the normal
direction respective to the boundary.
