Steady flows of Jeffrey–Hamel type from the half-plane into an infinite channel. 2. Linearization on a symmetric solution

AbstractThe steady flow of a Navier–Stokes fluid is analysed in a two-dimensional asymmetric unbounded domain Ω, having two outlets to infinity, namely a half-plane K and a semi-infinite channel Π−. Assuming that Ω differs from a symmetric domain Ω only by a small perturbation, we show the existence of a unique solution to the Navier–Stokes system in Ω. The solution is obtained as a perturbation of the symmetric solution and, at large distances in K, it takes the Jeffrey–Hamel form. Curiously, our results are valid only if the flux Φ, besides being small, is directed from the half-plane towards the semi-infinite channel, i.e. Φ is negative.The main ingredients in our proofs are estimates in weighted spaces with detached asymptotics and the study of a model problem resulting from the linearization around the symmetric solution which, for non-zero flux, leads, in contrast to the linearization around the zero solution, to the absence of compatibility conditions for the convective term and, for Φ