No-Existence of Global Smooth Solutions for the Three-Dimensional Navier-Stokes Equation

No-existence of global smooth solutions for the three-dimensional Navier-Stokes equation is proved. For plane Poiseuille flow, the unsteady laminar flow is decomposed into a time-averaged flow and a disturbance. At sufficiently high Reynolds numbers and large disturbances, the superposition effect of the time-averaged flow and the disturbances causes the viscous term (Laplacian term) of the composite flow to vanish at certain points in finite time, resulting in zero viscous energy loss. At such points, the nonlinear term amplifies the disturbance while there is no viscous role to damp the disturbance, which leads to increase of high-order derivatives of disturbances. Then, the velocity gradient becomes unbounded, producing velocity discontinuity. This result disproves the existence of global smooth solutions to the Navier-Stokes equation.