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

TL;DR

This paper proves the uniqueness and stability of Couette-Poiseuille flows in a 2D channel without restrictions on flux size, extending previous methods to demonstrate invertibility and stability under perturbations.

Contribution

It extends the analysis of Couette-Poiseuille flows by establishing their uniqueness and stability for arbitrary flux values without flow reversal restrictions.

Findings

Proved continuous invertibility of the linearized operator at non-reversing solutions.

Established local uniqueness and nonlinear stability under small forces.

Showed non-invertibility of the linearized operator when flow reversal occurs.

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.