Validity of Prandtl Expansion for Steady Compressible Navier-Stokes-Fourier Flows

TL;DR

This paper rigorously justifies the Prandtl layer expansion for steady compressible Navier-Stokes-Fourier flows, establishing uniform remainder estimates and confirming the expansion's validity in the inviscid limit.

Contribution

It develops a mathematical framework for uniform-in-v remainder estimates around Prandtl layers in compressible flows, validating the Prandtl expansion.

Findings

Proves the validity of Prandtl layer expansion in the steady compressible case.

Establishes uniform-in-v remainder estimates for the linearized equations.

Handles both velocity and thermal layers in the analysis.

Abstract

Assume no-slip boundary conditions for the velocity field and either insulated or Dirichlet boundary conditions for the temperature field in a steady compressible fluid. In the inviscid limit $\v \rightarrow 0$, we develop a mathematical framework for the uniform-in-$\v$ remainder estimate for the linear steady compressible Navier-Stokes-Fourier equations around a Prandtl layer profile with both velocity and thermal layers, which leads to the validity of the Prandtl layer expansion.