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.
A "distorted channel," C, is a two-dimensional domain consisting of two semi-infinite straight channels, C 1 and C 2 , connected smoothly by a bounded and regular domain, C 0 . One of the most intriguing and still unresolved problems in the mathematical theory of the Navier-Stokes equations is to prove (or disprove) for such domains the existence of a steady-state solution, that converges at large distances from C 0 to arbitrarily assigned Poiseuille flows in C 1 and C 2 [6,Chapter XIII]. Since these flows are characterized by their (equal) flux Φ through the generic cross-section of C, the problem can be reformulated by asking whether solutions exist for an arbitrary non-zero value of Φ that satisfy the aforementioned asymptotic conditions. 1 This question, originally posed by J. Leray to O.A. Ladyzhenskaya in the late 1950s (see [5,Remark 1.6]), is currently known as the "Leray problem."
The first contribution to Leray problem is due to Amick [1,2]. He looks for solutions whose velocity field u is of the form u = u * + v where u * is a smooth (given) extension of the Poiseuille flows to the whole C, and v is a “correction” satisfying a suitable perturbed nonlinear problem around u * . This construction produces existence, with v in a subspace of the Sobolev space H 1 (C), on condition that |Φ| is not too large [1]. A few years later, Ladyzhenskaya & Solonnikov [14] addressed, among others, Leray problem from a different perspective. Namely, they looked for solutions with u in a subspace of the local Sobolev space H 1 (C a,a+1 ), a ∈ R, where C a,a+1 is a cross-sectional slice of C located at a and of thickness 1; see (1.11). In this way, they were able to prove existence of solutions with u ∈ H 1 (C a,a+1 ), for arbitrary |Φ|. However, the asymptotic convergence of such u to the associated Poiseuille velocity fields in C i , i = 1, 2, is only guaranteed if |Φ| is not too large. 2The fact that both approaches presented in [1] and [14] require, for convergence to the Poiseuille flow, that the flux magnitude not be too large, raises the question of whether, for sufficiently large flux, there may exist corresponding bounded solutions different from the Poiseuille solutions, and whether it is precisely to the manifold of velocity fields of these other solutions that u might converge.
The question of the local uniqueness of Poiseuille flow -namely, absence of other solutions in a neighborhood of this flow-has been investigated by several authors in [17,18,21,22]. Postponing a detailed description of their results to a later point, we will simply state here that, in the general case, they all give a positive answer but on condition that the magnitude of the flux remains below a certain constant or else is above a suitable constant.
The main objective of this paper is to prove a rather comprehensive result on the local uniqueness of a class of parallel flows in a two-dimensional straight, infinite channel. Our results guarantee, in particular, that for arbitrary values of the flux Φ, the Poiseuille solution is, locally, the only possible one in both the functional settings of Amick [1] and Ladyzhenskaya & Solonnikov [14]. Moreover, such uniqueness property is proved to hold globally, in the subclass of solutions possessing suitable symmetries.
To present our results and the main ideas underlying our work, we begin to give the precise formulation of the problem. Let S = R × (-1, 1) denote the infinite channel, and consider the following boundaryvalue problem: (1.4)
The question we address is whether (1.3) is, in a suitable class, the only solution to (1.1). To this end, set Φ . = 2(A + C). We then look for solutions to (1.1) 1,2 in the form u(x, y) = u * (x, y) + v(x, y) and p(x, y) = p * (x, y) + q(x, y) ∀(x, y) ∈ S ,
where the pair (v, q) satisfies the following problem:
(1.8)
We emphasize that there are no convergence conditions like (1.1) 3 at the (infinite) inlet/outlet. The main objective is therefore to demonstrate that problem (1.8) admits only the trivial solution v ≡ ∇q ≡ 0, at least for “small” v, within a class of functions similar to those considered by Amick and Ladyzhenskaya & Solonnikov. A natural way of showing this is to prove that the linearized problem:
has only the trivial solution. This is ensured if the linear operator
referred to as the Couette-Poiseuille linearization, is, suitably defined, an isomorphism. In which case, a contraction-mapping argument, for example, will guarantee the desired uniqueness property in the appropriate classes.
The first rigorous contribution to the study of this problem is due to Rabier [17,18], in the case where u * is the Poiseuille flow (1.3)-(1.4) 1 .4 Specifically, in [17] the continuous invertibility of the linearization L for arbitrarily large flux is proved in the Sobolev spaces of functions enjoying appropriate symmetry properties. This result provides, as a byproduct, the local existence and uniqueness, for an arbitrary flux, of a symmetric
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