Well-posedness of thermal layer equations for inviscid compressible flows

TL;DR

This paper derives a semi-explicit solution formula for thermal boundary layers in inviscid compressible flows, analyzes their convergence to the inviscid Prandtl system, and studies their stability in three dimensions.

Contribution

It provides a new explicit solution formula for thermal boundary layers and establishes their convergence and stability properties in three-dimensional flows.

Findings

Explicit solution formula for thermal boundary layers

Convergence to the inviscid Prandtl system as initial temperature approaches a constant

Stability of the linearized system depends on the invariance of tangential velocity direction

Abstract

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.