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On the uniqueness and structural stability of Couette-Poiseuille flow in a channel for arbitrary values of the flux

Giovanni P. Galdi, Filippo Gazzola, Mikhail V. Korobkov, Xiao Ren, Gianmarco Sperone

TL;DR

The paper tackles the Leray problem for steady parallel flows in a 2D channel by focusing on Couette–Poiseuille baselines and showing that the linearized Couette–Poiseuille operator is an isomorphism without flux restrictions. It reformulates the problem via a stream-function approach and analyzes the non-homogeneous Orr–Sommerfeld equation for Fourier modes using a McLeod-inspired method, obtaining sharp a priori estimates. These results yield local nonlinear stability and global uniqueness in symmetric classes for arbitrary flux, and they demonstrate symmetry can lead to stronger conclusions. It also shows that invertibility can fail when flow reversal occurs, highlighting the necessity of the ABC-type conditions.

Abstract

We establish uniqueness and structural stability of a class of parallel flows in a 2D straight, infinite channel, under perturbations with either globally or locally bounded Dirichlet integrals. The significant feature of our result is that it does not require any restriction on the size of the flux characterizing the flow. Precisely, by extending and refining an approach initially introduced by J.B. McLeod, we demonstrate the continuous invertibility of the linearized operator at a generic Couette-Poiseuille solution that does not exhibit flow reversal. We then deduce local uniqueness of these solutions as well as their nonlinear structural stability under small external forces. Moreover, we prove the uniqueness of certain class of Couette-Poiseuille solutions ``in the large," within the set of solutions possessing natural symmetry. Finally, we bring an example showing that, in general, if the flow reversal assumption is violated, the linearized operator is no longer invertible.

On the uniqueness and structural stability of Couette-Poiseuille flow in a channel for arbitrary values of the flux

TL;DR

The paper tackles the Leray problem for steady parallel flows in a 2D channel by focusing on Couette–Poiseuille baselines and showing that the linearized Couette–Poiseuille operator is an isomorphism without flux restrictions. It reformulates the problem via a stream-function approach and analyzes the non-homogeneous Orr–Sommerfeld equation for Fourier modes using a McLeod-inspired method, obtaining sharp a priori estimates. These results yield local nonlinear stability and global uniqueness in symmetric classes for arbitrary flux, and they demonstrate symmetry can lead to stronger conclusions. It also shows that invertibility can fail when flow reversal occurs, highlighting the necessity of the ABC-type conditions.

Abstract

We establish uniqueness and structural stability of a class of parallel flows in a 2D straight, infinite channel, under perturbations with either globally or locally bounded Dirichlet integrals. The significant feature of our result is that it does not require any restriction on the size of the flux characterizing the flow. Precisely, by extending and refining an approach initially introduced by J.B. McLeod, we demonstrate the continuous invertibility of the linearized operator at a generic Couette-Poiseuille solution that does not exhibit flow reversal. We then deduce local uniqueness of these solutions as well as their nonlinear structural stability under small external forces. Moreover, we prove the uniqueness of certain class of Couette-Poiseuille solutions ``in the large," within the set of solutions possessing natural symmetry. Finally, we bring an example showing that, in general, if the flow reversal assumption is violated, the linearized operator is no longer invertible.
Paper Structure (9 sections, 11 theorems, 148 equations)

This paper contains 9 sections, 11 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Reformulation of the problem in terms of the stream function
  3. A non-homogeneous Orr-Sommerfeld equation
  4. Uniqueness and structural stability in the $L^2$-setting
  5. Uniqueness and structural stability in the locally $L^2$-setting
  6. Symmetric flows
  7. Uniqueness of symmetric solutions to the nonlinear problem
  8. Non-invertibility of the linearization around some symmetric flows
  9. Some historical facts

Key Result

Proposition 1

Assume ABC, and let $\mathcal{F}$ be as in parabolasign. Then for any $\xi \neq 0$ and $h_\xi \in H^{-1}((-1,1);\mathbb{C})$, there exists a unique solution $\varphi_\xi \in H^3((-1,1); \mathbb{C})\cap H^2_0((-1,1); \mathbb{C})$ to OS satisfying

Theorems & Definitions (25)

  • Remark 1
  • Remark 2
  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • ...and 15 more