Energy Flux Criteria and Critical Besov Regularity for the 3D Navier–Stokes Equations

We investigate how the flow of energy across different scales affects the regularity of solutions to the three-dimensional Navier–Stokes equations, which describe the motion of viscous incompressible fluids. Using tools from harmonic analysis (the Littlewood–Paley decomposition and Bony’s paraproduct), we give a precise definition of the energy flux from one scale to the next and prove a new criterion for when a solution stays smooth. Our main result shows that if a weak solution belongs to a certain critical Besov space and, in addition, the weighted cumulative energy flux decays faster than the critical Onsager rate, then the solution can be extended smoothly past any potential blow‑up time. The proof uses a weighted dyadic estimate, Bernstein’s inequality, and an interpolation argument that leads to a Serrin‑type condition. The borderline case corresponds exactly to the Onsager threshold for energy dissipation; any improvement beyond that threshold guarantees regularity. This establishes a rigorous link between the turbulence energy cascade and the prevention of singularities. The criterion is physically transparent, quantifies how much kinetic energy is transferred to very small scales, and provides a new diagnostic for numerical simulations. Our work contributes to the long‑standing effort to understand the global regularity problem for the Navier–Stokes equations, one of the Clay Millennium Prize Problems.