Asymptotic analysis of a two--dimensional coupled problem for compressible viscous flows

We consider a two--dimensional coupled transmission problem with the conservation laws for compressible viscous flows, where in a subdomain $\Omega_1$ of the flow--field domain $\Omega$ the coefficients modelling the viscosity and heat conductivity are set equal to a small parameter $\varepsilon>0$. The viscous/viscous coupled problem, say $P_\varepsilon$, is equipped with specific boundary conditions and natural transmission conditions at the artificial interface $\Gamma$ separating $\Omega_1$ and $\Omega \setminus \Omega_1$. Here we choose $\Gamma $ to be a line segment. The solution of $P_\varepsilon$ can be viewed as a candidate for the approximation of the solution of the real physical problem for which the dissipative terms are strongly dominated by the convective part in $\Omega_1$. With respect to the norm of uniform convergence, $P_\varepsilon$ is in general a singular perturbation problem. Following the Vishik--Ljusternik method, we investigate here the boundary layer phenomenon at $\Gamma$. We represent the solution of $P_\varepsilon$ as an asymptotic expansion of order zero, including a boundary layer correction. We can show that the first term of the regular series satisfies a reduced problem, say $P_0$, which includes the inviscid/viscous conservation laws, the same initial conditions as $P_\varepsilon$, specific inviscid/viscous boundary conditions, and transmission conditions expressing the continuity of the normal flux at $\Gamma$. A detailed analysis of the problem for the vector--valued boundary layer correction indicates whether additional local continuity conditions at $\Gamma$ are necessary for $P_0$, defining herewith the reduced coupled problem completely. In addition, the solution of $P_0$ (which can be computed numerically) plus the boundary layer correction at $\Gamma$ (if any) provides an approximation of the solution of $P_\varepsilon$ and, hence, of the physical solution as well. In our asymptotic analysis we mainly use formal arguments, but we are able to develop a rigorous analysis for the separate problem defining the correctors. Numerical results are in agreement with our asymptotic analysis.