Uniqueness of weak solutions to the three-dimensional Navier–Stokes equations

Determine whether Leray weak solutions of the three-dimensional incompressible Navier–Stokes initial value problem on R^3 are unique. Concretely, for the system ∂u/∂t − μΔu + (u · ∇)u + ∇p = 0, div u = 0 in R^3 × (0, ∞) with divergence-free initial data and the usual finite-energy/suitable-growth conditions ensuring existence of Leray weak solutions, establish or refute uniqueness of such weak solutions.

Background

The paper reviews classical results on the Navier–Stokes equations, noting that Leray established the existence of weak solutions in three dimensions with suitable growth properties. However, despite global existence, the uniqueness of these weak solutions has not been established.

This issue is distinct from the global regularity problem addressed by the author via a limiting procedure from parabolic Lamé equations; the open question here concerns the uniqueness of weak (Leray) solutions without additional regularity or smallness assumptions.

References

Leray in showed that the Navier-Stokes equations ((i)) in three space dimensions always have a weak solution $(\boldsymbol{u},p)$ with suitable growth properties, but the uniqueness of weak solutions of the Navier-Stokes equations is not known.

— Existence of smooth solutions of the Navier-Stokes equations in three-dimensional Euclidean space  (2507.18063 - Liu, 24 Jul 2025) in Introduction (Section 1)

Leray and Hopf constructed global finite-energy weak solutions, but their regularity and uniqueness in three dimensions remain open .

— A Critical Chemin--Lerner Regularity Criterion via One Velocity Component for the Three-Dimensional Navier--Stokes Equations  (2609.03877 - Guo et al., 3 Sep 2026) in Section 1, Introduction and main results

However, whether such Leray-Hopf weak solutions are unique remains one of the most challenging open problems in the analysis of nonlinear partial differential equations. Despite significant progress over the past decades, the problem remains open for both bounded and unbounded domains.

— Divergence-Free Approximation in Sobolev and Lebesgue Spaces on General Unbounded Domains with Applications to Energy Equality in Fluid Dynamics  (2609.02769 - Khan et al., 2 Sep 2026) in Section 1, subsection “The Navier-Stokes equations”