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}.
Categories: math.AP