  In this article we prove, choosing an appropriately weighted $L_2$-energy
equivalent to the classical energy, that the plane Couette and Poiseuille flows
are nonlinearly stable with respect to streamwise perturbations for any
Reynolds number. In this case the coefficient of time-decay of the energy is
$\pi^2/(2 {\rm Re})$, and it is a bound from above of the time-decay of
streamwise perturbations of linearized equations. We also prove that the plane
Couette and Poiseuille flows are nonlinearly stable if the Reynolds number is
less then ${\rm Re}_{Orr}/\sin \varphi$ when the perturbation is a tilted
perturbation, i.e. 2D perturbations with wave vector which forms an angle
$\varphi \in [0, \pi/2]$ with the direction ${\bf i}$ of the motion. ${\rm
Re}_{Orr}$ is the Orr (1907) critical Reynolds number for spanwise
perturbations which, for the Couette flow is ${\rm Re}_{Orr}=177.22$ and for
the Poiseuille flow is ${\rm Re}_{Orr}=175.31$.
  In particular these results improve those obtained by Joseph (1966), who
found for streamwise perturbations a critical nonlinear value of $82.6$ in the
Couette case, and those obtained by Joseph and Carmi (1969), who found the
value $99.1$ for plane Poiseuille flow for streamwise perturbations. The
results we obtain here are, for any angle, in a good agreement with the
experiments (see Prigent et al. 2003) and the numerical simulations (see
Barckley and Tuckerman 2005, 2007).
