  We study the nonlinear stability of plane Couette and Poiseuille flows with
the Lyapunov second method by using the classical L2-energy. We prove that the
streamwise perturbations are L2-energy stable for any Reynolds number. This
contradicts the results of Joseph [10], Joseph and Carmi [12] and Busse [4],
and allows us to prove that the critical nonlinear Reynolds numbers are
obtained along two-dimensional perturbations, the spanwise perturbations, as
Orr [16] had supposed. This conclusion combined with recent results by
Falsaperla et al. [8] on the stability with respect to tilted rolls, provides a
possible solution to the Couette-Sommerfeld paradox.
