Non-Existence of Global Smooth Solutions to the Navier–Stokes Equations in a 3D Periodic Domain with Prescribed Body Force
Abstract
We rigorously prove that global smooth solutions do not exist for the three-dimensional incompressible Navier–Stokes equations defined on a periodic rectangular domain $\Omega$ with a prescribed smooth body force. The smooth initial data is obtained 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 the sum of the viscous term and the non-conservative 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 the non conservative body force. Nonlinear interactions amplify disturbances and lead to a local cancellation of the sum of the mean flow viscous term and the non conservative body force with the disturbance viscous term at a finite critical time $t^*>0$ and an interior point $\boldsymbol{x}^*\in\Omega$. This cancellation leads to the local mechanical energy gradient along the streamline being zero when the time derivative is zero, which by EVMP implies $\boldsymbol{u}(\boldsymbol{x}^*,t^*)=0$, contradicting the existing non‑vanishing mean 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.…
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