Persistence of Steady 3D Euler Solutions for 3D Navier-Stokes Equations

In the classical plane Couette flow, certain 3D steady solution (the so-called lower branch state) of the Navier-Stokes equations has a nontrivial limit as the Reynolds number approaches infinity [8]. The limit is a shear of the form (U(y, z), 0, 0) in velocity variables. On the other hand, all the shears of this form are solutions of the corresponding 3D Euler equations. This note derives a necessary condition for such a shear to be a limit shear. The condition is R ∆Uf(U)dydz = 0 for any function f satisfying certain boundary condition. Similar conditions are also derived for plane Poiseuille flow and pipe Poiseuille flow, which correspond to similar limit shears as revealed in [7] [6].