Disproving the Existence of Global Smooth Solutions to the Navier-Stokes Equation

In laminar flow with disturbances, the instantaneous flow is decomposed into the mean flow and the disturbances. At a sufficiently large Reynolds number, the nonlinear evolution alters the mean flow profile. As the amplitude of the disturbances grows, at certain points, the viscous term of the disturbances cancels out the viscous term of the mean flow, causing the viscous term of the instantaneous flow to tend towards zero. Due to the mismatch between the energy loss and the velocity magnitude at these points, they become physical singularities of the Navier-Stokes equation. The instability of these singularities leads to velocity discontinuities, where the norm of the velocity gradient tends to infinity. This proves that there is no smooth solution to the Navier-Stokes equation in the global domain. This paper takes the plane Poiseuille flow as an example to demonstrate the physical logic and mathematical derivation process of the proof.