Zero-State Membership of the Global Attractor

Determine whether the global attractor of the two-dimensional Navier–Stokes equations contains the zero state for a given time-independent body force.

Background

The paper compares its finite-time viscosity-identifiability criteria with earlier criteria formulated on the global attractor. Those earlier criteria require information about whether the zero solution belongs to the global attractor associated with a given time-independent forcing.

The authors explicitly identify this membership question as unresolved. Its resolution would clarify the scope of attractor-based identifiability theories for the two-dimensional Navier–Stokes equations.

References

However, at the moment, whether or not the global attractor contains the zero state for a given body time-independent body force remains a longstanding open problem.

— Excitation and Identifiability in the 2D Navier-Stokes equations  (2609.31230 - Martinez et al., 25 Sep 2026) in Section 1, subsection “Data assimilation”