On the Stability of Shear Flows in Bounded Channels, I: Monotonic Shear Flows

We discuss some of our recent work on the linear and nonlinear stability of shear flows as solutions of the 2D Euler equations in the bounded channel T×[0,1]. More precisely, we consider shear flows u=(b(y),0) given by smooth functions b:[0,1]→R. We prove linear inviscid damping and linear stability provided that b is strictly increasing and a suitable spectral condition involving the function b is satisfied. Then we show that this can be extended to full nonlinear inviscid damping and asymptotic nonlinear stability, provided that b is linear outside a compact subset of the interval (0, 1) (to avoid boundary contributions which are not compatible with inviscid damping) and the vorticity is smooth in a Gevrey space. In the second article in this series we will discuss the case of non-monotonic shear flows b with non-degenerate critical points (like the classical Poiseuille flow b:[-1,1]→R, b(y)=y²). The situation here is different, as nonlinear stability is a major open problem. We will prove a new result in the linear case, involving polynomial decay of the associated stream function.