  In this paper, we study the nonlinear stability for the 3-D plane Poiseuille
flow $(1-y^2,0,0)$ at high Reynolds number $Re$ in a finite channel
$\mathbb{T}\times [-1,1 ]\times \mathbb{T}$ with non-slip boundary condition.
We prove that if the initial velocity $v_0$ satisfies
$\|v_0-(1-y^2,0,0)\|_{H^{4}}\leq c_0 Re^{-\frac{7}{4}}$ for some $c_0>0$
independent of $Re$, then the solution of 3-D Naiver-Stokes equations is global
in time and does not transit away from the plane Poiseuille flow. To our
knowledge, this is the first nonlinear stability result for the 3-D plane
Poiseuille flow and the transition threshold is accordant with the numerical
result by Lundbladh et al. \cite{LHR}.
