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Structural stability of boundary layers in the entire subsonic regime

Shengxin Li, Tong Yang, Zhu Zhang

TL;DR

This work delivers a rigorous, unified stability theory for boundary-layer profiles in the 2D steady compressible Navier–Stokes system across the entire subsonic regime $m\in(0,1)$. The authors decompose the problem into Fourier modes in the tangential direction and develop a robust quasi-compressible–Stokes iteration to solve the linearized system, then reduce nonzero modes to a compressible Orr–Sommerfeld equation OS_CNS solved via Rayleigh–Airy iterations in the low/middle frequencies and by energy methods at high frequencies. Boundary-layer corrections are constructed across low, middle, and high frequency regimes to enforce no-slip conditions, and nonlinear stability is achieved through a KN19-inspired modified linear system with careful high-order estimates, culminating in a nonlinear stability and a low Mach limit with incompressible boundary-layer leading order $(1,U_s(Y),0)$. The results yield uniform-in-$m$ estimates, establish the first low-Mach limit result in the presence of Prandtl boundary layers for the Navier–Stokes system, and deepen the understanding of density–velocity interactions in compressible boundary layers. These findings have implications for rigorous boundary-layer analysis in compressible flows under Sobolev regularity and subsonic regimes.

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.

Structural stability of boundary layers in the entire subsonic regime

TL;DR

This work delivers a rigorous, unified stability theory for boundary-layer profiles in the 2D steady compressible Navier–Stokes system across the entire subsonic regime . The authors decompose the problem into Fourier modes in the tangential direction and develop a robust quasi-compressible–Stokes iteration to solve the linearized system, then reduce nonzero modes to a compressible Orr–Sommerfeld equation OS_CNS solved via Rayleigh–Airy iterations in the low/middle frequencies and by energy methods at high frequencies. Boundary-layer corrections are constructed across low, middle, and high frequency regimes to enforce no-slip conditions, and nonlinear stability is achieved through a KN19-inspired modified linear system with careful high-order estimates, culminating in a nonlinear stability and a low Mach limit with incompressible boundary-layer leading order . The results yield uniform-in- estimates, establish the first low-Mach limit result in the presence of Prandtl boundary layers for the Navier–Stokes system, and deepen the understanding of density–velocity interactions in compressible boundary layers. These findings have implications for rigorous boundary-layer analysis in compressible flows under Sobolev regularity and subsonic regimes.

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 . 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.

Paper Structure

This paper contains 35 sections, 44 theorems, 463 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Strategy of Proof
  5. Linear stability of zero mode
  6. Compressible Orr-Sommerfeld equation
  7. Derivation of an Orr-Sommerfeld approximation
  8. Some estimates on Rayleigh and Airy solutions
  9. Rayleigh-Airy iteration
  10. Solvability at high frequencies
  11. Inhomogeneous quasi-compressible system
  12. Homogeneous quasi-compressible solutions
  13. Low-frequency regime $\hat{n}\ll \nu^{-\frac{3}{4}}$
  14. Slow modes.
  15. Fast mode
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let the Mach number $m\in (0,1)$. Suppose that $U_s$ satisfies structural conditions 2.1-2.2. Then there exists a positive constant $L_0$, such that for any $L\in (0, L_0)$ and $0<\nu\ll1$, if $\|(F_{{\rm ext},1}, F_{{\rm ext},2})\|_{w}\leq \nu^{\frac{9}{8}+}$, the system 1.5 admits a unique solutio Moreover, the solution satisfies the following zero-mass condition

Theorems & Definitions (81)

  • Theorem 1.1: Structural Stablity
  • Remark 1.1
  • Theorem 1.2: Low Mach number limit
  • Remark 1.2
  • Theorem 2.1
  • Remark 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 71 more