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

Abstract

We construct a nonzero divergence-free initial velocity on $\mathbb{R}^3$ 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 $L^{\mathfrak q}$ with $1\leq \mathfrak q<3$, but not to $L^3$. All the solutions attain this datum strongly in $L^2$, are smooth at positive times, and satisfy both the global and local energy equalities through time zero. The same family belongs simultaneously to every $L^{\mathfrak p}_tL^{\mathfrak q}_x$ with $1\leq \mathfrak p\leq\infty$, $2\leq \mathfrak q\leq\infty$, and $2/\mathfrak p+3/\mathfrak q>1$. The same solutions also belong to the critical Lorentz space $L^\infty_tL^{3,\infty}_x$. 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 $L^2$ 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.