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Non-Uniqueness of Leray-Hopf Solutions to the Unforced Three-Dimensional Navier-Stokes Equations

Primarily AI-generated textHuman understanding: some partsmath.AP — Analysis of PDEs

Contributed by Tristan Buckmaster, Matei Coiculescu ↗, Chen Li

Version 1 / Nov 06, 2026 / CC BY 4.0

Abstract

We prove that the incompressible Navier--Stokes equations in R3\R^3, without external force, admit a continuum of distinct suitable weak solutions in the Leray--Hopf class with the same initial datum. The datum is compactly supported, axisymmetric without swirl, and smooth away from the origin; it lies in L3,∞L^{3,\infty} and in LqL^q for every q<3q<3, but not in L3L^3. The solutions are axisymmetric without swirl and smooth at positive times, and they satisfy the local and global energy equalities. Any two of them separate in L2L^2 at the rate t1/4+σt^{1/4+\sigma}, for a fixed 0<σ<1/40<\sigma<1/4. The proof follows the program of Jia and \v Sver\'ak, with a discretely self-similar background in place of a self-similar one. The background is an exact unforced solution built from a train of thin vortex rings, which we obtain by lifting a periodic row of planar vortices with Gaussian stream function. We construct a growing Floquet mode of its linearization directly, from the angular-two instability of the planar vortex, and conclude by a nonlinear correction and a localization of the datum.

Author comments: Lean formalization forthcoming at the public repository at https://github.com/mcoiculescu/unforced-leray-hopf-public Exposition will be improved in future versions. We used an adaptation of Scott Armstrong's skill files for the Lean formalization process.

Provenance statement

The broad steps of the construction have been understood by the authors. It is essentially the implementation of the Jia-Sverak program for a discretely self-similar "background" flow, which is itself explicitly constructed.

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Version history

  1. v1 · 80 pagesSubmitted by Matei CoiculescuLean formalization forthcoming at the public repository at https://github.com/mcoiculescu/unforced-leray-hopf-public Exposition will be improved in future versions. We used an adaptation of Scott Armstrong's skill files for the Lean formalization process.CurrentNov 06, 2026