In this paper, we study the nonlinear stability for the 3-D plane Poiseuille
flow (1βy2,0,0) at high Reynolds number Re in a finite channel
TΓ[β1,1]ΓT with non-slip boundary condition.
We prove that if the initial velocity v0β satisfies
β₯v0ββ(1βy2,0,0)β₯H25β,2ββ€c0βReβ47β for some
c0β>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}