Navier–Stokes non-uniqueness from almost identical initial conditions

Establish whether the Navier–Stokes equations admit distinct smooth global solutions for all t > 0 corresponding to initial conditions U1(r) and U2(r) that satisfy |U1(r) − U2(r)| ≤ δ for arbitrarily small δ, including the limit δ → 0.

Background

The paper presents clean numerical simulations (CNS) of a two-dimensional forced Navier–Stokes Kolmogorov flow demonstrating divergent trajectories and statistics from initial conditions that differ by perturbations as small as 10{-40}. The authors emphasize that CNS reduces numerical noise to negligible levels over a long finite time interval, enabling high-fidelity comparisons of solutions.

They design initial data U1 with rotation and translation symmetry and U2 = U1 + δ U0 with only translation symmetry, and show that the resulting flows differ in symmetry, temporal behavior of dissipation, and statistical measures, even when δ is extremely small. Motivated by these observations, they propose a general conjecture asserting non-uniqueness of smooth global Navier–Stokes solutions arising from almost identical initial data.

References

Our CNS results mentioned above highly suggest the following conjecture: The Navier-Stokes equations admit distinct smooth global solutions for all $t>0$ from almost the same initial conditions $U_1({\bf r})$ and $U_2({\bf r})$ with $\left| U_1({\bf r})-U_2({\bf r})\right|\leq \delta$ for arbitrarily small $\delta$ (or as $\delta \to 0$).

— Non-uniqueness of smooth solutions of the Navier-Stokes equations from almost the same initial conditions  (2602.12666 - Liao et al., 13 Feb 2026) in Section: Concluding remarks and discussions