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 $1\le \mathfrak{q}<3$, 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 $L_t^{\mathfrak{p}}{L}_x^{\mathfrak{q}}$ with $1\le \mathfrak{p}\le \infty$, $2\le \mathfrak{q}\le \infty$, and $2/\mathfrak{p}+3/\mathfrak{q}>1$. The same solutions also belong to the critical Lorentz space $L_t^{\infty }{L}_x^{3,\infty }$. 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.

arXiv comment113 pages, 2 figures (4 panels)