On the Stability of Laminar Flows Between Plates

Consider a two-dimensional laminar flow between two plates, so that (x(1), x(2)) is an element of R x [-1, 1], given by v(x(1), x(2)) = (U(x(2)), 0), where U is an element of C-4([-1, 1]) satisfies U' not equal 0 in [-1, 1]. We prove that the flow is linearly stable in the large Reynolds number limit, in two different cases: sup(x is an element of[- 1,1]) vertical bar U '' (x)vertical bar+sup(x is an element of[- 1,1]) vertical bar U ''' (x)vertical bar << min(x is an element of[- 1,1]) vertical bar U' (x)vertical bar (nearly Couette flows), U '' not equal 0 in [-1, 1]. We assume either no-slip or fixed traction force (Navier-slip) conditions on the plates, and an arbitrary large (but much smaller than the Reynolds number) period in the x(1) direction.