Structural stability of boundary layers in the entire subsonic regime

TL;DR

This paper proves the structural stability of boundary layer shear flows in the steady compressible Navier-Stokes equations across all subsonic Mach numbers, also addressing the low Mach number limit with boundary layers.

Contribution

It establishes the first uniform stability results for boundary layers in the entire subsonic regime and introduces new analytical techniques for the low Mach number limit.

Findings

Uniform estimates across subsonic Mach numbers

First results on low Mach number limit with boundary layers

Identification of cancellations in higher-order estimates

Abstract

Despite the physical importance, there are limited mathematical theories for the compressible Navier-Stokes equations with strong boundary layers. This is mainly due to the absence of a stream function structure, unlike the extensively studied incompressible fluid dynamics in two dimensions. This paper aims to establish the structural stability of boundary layer profiles in the form of shear flow for the two-dimensional steady compressible Navier-Stokes equations. Our estimates are uniform across the entire subsonic regime, where the Mach number $m\in (0,1)$. As a byproduct, we provide the first result concerning the low Mach number limit in the presence of Prandtl boundary layers. The proof relies on the quasi-compressible-Stokes iteration introduced in [38], along with a subtle analysis of the interplay between density and velocity variables in different frequency regimes, and the identification of cancellations in higher-order estimates.