On stability of steady-state plane-parallel shearing flows in a homogeneous in density ideal incompressible fluid

A problem on linear stability of stationary plane-parallel shearing flows in a homogeneous in density inviscid incompressible fluid between two immovable impermeable solid parallel infinite plates is studied. With the use of the direct Lyapunov method it is shown that all sufficient conditions (by Rayleigh, Fjørtoft, Arnol'd) known to this moment for stability of these flows with respect to small plane perturbations are the necessary ones as well. An a priori lower estimate is constructed; the estimate displays exponential in time growth of the considered perturbations if these conditions are not affected. An analytical example of steady-state plane-parallel shearing flows and superimposed small plane perturbations growing in time in accordance with the constructed estimate is given.