Non-Existence of Global Smooth Solutions to the 3D Navier–Stokes Equations in a Periodic Domain with Prescribed Body Force

We rigorously disprove the existence of global smooth solutions to the three‑dimensional incompressible Navier–Stokes equations in a periodic rectangular cuboid domain $\Omega$ subject to the prescribed smooth body force. The smooth initial data of the flow field is derived from a two-dimensional stationary exact solution. The analysis is grounded in Sobolev space regularity, the decomposition of velocity into a time‑averaged mean flow and a disturbance flow, the local vanishing of sum of the viscous term and the body force, and the Energy–Velocity Monotonicity Principle (EVMP). When the Reynolds number exceeds the critical value for turbulent transition, the nonlinear convective term dominates over the viscous term and external force term, nonlinear interactions amplify disturbances, leading to local cancellation of the sum of the mean flow viscous term and the body force with the disturbance viscous term at a finite critical time $t^*>0$ and interior point $\boldsymbol{x}^*\in\Omega$. This cancellation leads to the local mechanical energy gradient along the streamline being zero when the time derivativer is zero, which by EVMP requires $|\boldsymbol{u}(\boldsymbol{x}^*,t^*)|=0$, contradicting the existing non‑vanishing velocity. The contradiction generates a finite‑time regularity singularity, under which the velocity gradient $L^\infty$‑norm diverges. This violates the Sobolev embedding condition required for global smoothness of solutions. This study resolves the problem statement (D) in Fefferman (2006).