STUDY OF BOUNDARY LAYERS IN COMPRESSIBLE NON-ISENTROPIC FLOWS

In this note, we review our recent study on boundary layer problems in compressible non-isentropic flows with non-slip boundary condition for the velocity, in the small viscosity and heat conductivity limit. By multi-scale analysis, we derive the problems of viscous layer profiles and thermal layer profiles for different scales of viscosity and heat conductivity, from which we obtain the interaction mechanism of viscous layers and thermal layers. Then, when the viscosity goes to zero slower than or at the same rate as the heat conductivity, we give a well-posedness result of the twodimensional viscous layer problem, which is the Prandtl type equations coupled with a degenerated parabolic equation, in the class of tangential velocity being strictly monotonic in the normal variable. Last, when the viscosity goes to zero faster than the heat conductivity, we study the stability of the thermal layer problem at a shear flow in two or three space variables, which is an inviscid Prandtl type equations coupled with a degenerated parabolic equation.