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A semi-explicit formula of solution to the boundary layer system for thermallayer derived from the compressible Navier-Stokes equations with the non-slipboundary condition when the viscosity coefficients vanish is given, inparticular in three space dimension. In contrast to the inviscid Prandtl systemstudied by [7] in two space dimension, the main difficulty comes from thecoupling of the velocity field and the temperature field through a degenerateparabolic equation. The convergence of these boundary layer equations to theinviscid Prandtl system is justified when the initial temperature goes to aconstant. Moreover, the time asymptotic stability of the linearized systemaround a shear flow is given, and in particular, it shows that in three spacedimension, the asymptotic stability depends on whether the direction oftangential velocity field of the shear flow is invariant in the normaldirection respective to the boundary.

Well-posedness of thermal layer equations for inviscid compressible flows