A Viscous Cascade Construction of Nonunique Leray--Hopf Solutions to the Unforced 3D Navier--Stokes Equations

We construct a nonzero divergence-free initial velocity on R³ and a continuum of pairwise distinct suitable Leray--Hopf solutions of the unforced three-dimensional Navier--Stokes equations on a common time interval. The initial datum is axisymmetric, smooth away from the origin, and belongs to every Lq with 1≤ q<3, but not to L³. All the solutions attain this datum strongly in L², are smooth at positive times, and satisfy both the global and local energy equalities through time zero. The same family belongs simultaneously to every LᵖₜLqₓ with 1≤ p≤∞, 2≤ q≤∞, and 2/ p+3/ q>1. The same solutions also belong to the critical Lorentz space L^∞ₜL3,∞ₓ. This endpoint is sharp for uniqueness in the Leray--Hopf class within the Lorentz scale. The proof is entirely analytic and uses a cascade through geometrically separated Gaussian swirls. We establish growth for the time-dependent viscous linearization and determine the sign of its complete quadratic stress, including the interaction of the reference velocity with the second variation. The integrated stress generates a controlled perturbation at the next larger spatial scale. At each finite cutoff, a prescribed terminal coordinate selects an initial perturbation whose physical L² norm tends to zero as the cutoff is removed. Uniform energy, moment, and transfer estimates yield the common strong initial trace while preserving a nonzero observable at a fixed positive time.