Stability of Boundary Layers for the Nonisentropic Compressible Circularly Symmetric 2D Flow

Abstract

In this paper, we study the asymptotic behavior of circularly symmetric solutions to the initial boundary value problem of the nonisentropic compressible Navier--Stokes equations in a two-dimensional exterior domain with impermeable boundary conditions when the viscosities and the heat conduction coefficient tend to zero at the same rate. By multiscale analysis, we obtain that away from the boundary the nonisentropic compressible viscous flow can be approximated by the corresponding inviscid flow, and near the boundary there are boundary layers for the angular velocity, density, and temperature in the leading order expansions of solutions, while the radial velocity and pressure do not have boundary layers in the leading order. The boundary layers of velocity and temperature are described by a nonlinear parabolic system. We prove the stability of boundary layers and rigorously justify the asymptotic behavior of solutions in the $L^{\infty}$ space for the small viscosities and heat-conduction limit in the L...