Explicit nontrivial globally backward-extendable data in two dimensions

Construct an explicit initial condition \(u_0\in\mathcal{G}\), for the two-dimensional unforced Navier–Stokes equations, that is globally backward-extendable and is not a degenerate trigonometric polynomial or a trivial infinite-support variation.

Background

For the two-dimensional Navier–Stokes equations with zero forcing, the set $\mathcal{G}$ of globally backward-extendable data is dense in $H$, as established through a nonconstructive density argument. Degenerate trigonometric polynomials provide explicit examples.

The paper states that it remains unclear how to produce an explicit example outside those degenerate or trivial classes. Such a construction would complement the density theorem with a concrete nontrivial globally extendable solution.