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.