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 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.
Related
- Non-Uniqueness of Weak Solutions to the Unforced Two-Dimensional Euler Equations with Vorticity in L1∩ Lp, Part I: A Compactness Criterion and the Linear Analysis
- Uniqueness of Leray-Hopf weak solutions to the two-dimensional viscous Primitive Equations
- Global Well-Posedness and Large Time Asymptotic Behavior of Strong Solutions to the Compressible Viscous Non-Resistive MHD Equations with Vacuum
- Self-Similar Solutions of the 2D Incompressible Euler Equations via a Lagrangian Construction
- Bore wave solutions to the viscous Saint-Venant system with capillarity
BibTeX
@misc{hou2026viscous,
title = {{A Viscous Cascade Construction of Nonunique Leray--Hopf Solutions to the Unforced 3D Navier--Stokes Equations}},
author = {Thomas Y. Hou and Xiang Qin and Peicong Song and Wenhan Zhang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06117},
eprint = {2611.06117},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06117},
}
RIS
TY - UNPB
TI - A Viscous Cascade Construction of Nonunique Leray--Hopf Solutions to the Unforced 3D Navier--Stokes Equations
AU - Thomas Y. Hou
AU - Xiang Qin
AU - Peicong Song
AU - Wenhan Zhang
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06117
UR - https://arxiv.org/abs/2611.06117
AB - We construct a nonzero divergence-free initial velocity on 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^q with 1≤ 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^p_tL^q_x with 1≤ p≤∞, 2≤ q≤∞, and 2/p+3/q>1. The same solutions also belong to the critical Lorentz space L^∞_tL^3,∞_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.
ER -