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

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.