Strong determining modes for the three-dimensional Navier–Stokes equations

Determine whether the strong pointwise-in-time determining-modes property holds for the three-dimensional classical Navier–Stokes equations, namely, whether equality of sufficiently many low Stokes-mode projections for two complete trajectories on the global attractor at every time forces equality of the complete trajectories.

Background

The paper establishes a strong determining-modes property for complete trajectories on the global attractor of its three-dimensional Navier–Stokes-alpha model with nonlinear filtering. In this property, equality of the projections onto a sufficiently large finite-dimensional space of Stokes modes at every time implies equality of the full trajectories at every time.

The authors contrast this result with the classical three-dimensional Navier–Stokes equations, for which the corresponding strong determining-modes property is explicitly identified as unresolved. The nonlinear alpha filtering supplies the additional regularity and damping estimates needed for the result in the model studied here, but does not resolve the classical problem.