A geometric characterization of steady laminar flow

We study the steady states of the Euler equations on the periodic channel or annulus. We show that if these flows are laminar (layered by closed non-contractible streamlines which foliate the domain), then they must be either parallel or circular flows. We also show that a large subset of these shear flows are isolated from non-shear stationary states. For Poiseuille flow, $(v(y),0)=(y^2,0)$, our result shows that all stationary solutions in a sufficiently small $C^2$ neighborhood are shear flows. We then show that if $v(y)=y^n$ with $n \geq 1$, then in any $C^{n-}$ neighborhood, there exist smooth non-shear steady states, traveling waves, and quasiperiodic solutions of any number of non-commensurate frequencies. This proves the rigidity near Poiseuille is sharp. Finally, we prove that on general compact doubly connected domains, laminar steady Euler flows with constant velocity on the boundary must also be either parallel or circular, and the domain a periodic channel or an annulus. This shows that laminar free boundary Euler solutions must have Euclidean symmetry.