We introduce a scale-covariant reformulation of the three-dimensional incompressible Navier–Stokes equations based on a logarithmic radial coordinate (s = ln r), which maps multiplicative scaling to translation on an infinite line. Under this transformation, dilation-invariant dynamics are governed by a one-dimensional transport–diffusion equation for a scale-energy density E (s, t), generated by a scale-covariant operator (OSCO) with bounded transport velocity and positive scale viscosity. Within this framework, potential finite-time singularities in Euclidean space correspond to continuous energy flux toward the horizon (s → -∞), rather than local blow-up. We establish a conserved energy ledger identity on the scale line and prove global smoothness for small initial data in a scale-weighted Sobolev space H¹Onu. For arbitrary smooth data, we obtain conditional global regularity assuming uniform control of angular concentration on ℝ², isolating vortex stretching as the sole obstruction within the scale-covariant dynamics. The approach unifies elements of Leray self-similarity, Mellin convolution, and multiresolution analysis, and provides a geometric reinterpretation of the Navier–Stokes cascade that separates coordinate artifacts from physical singularities. Limitations and directions toward removing the angular regularity condition via Littlewood–Paley analysis are discussed.
Casey Riley (Tue,) studied this question.