Explicit non-extendable trigonometric polynomials

Construct an explicit trigonometric polynomial that is non-extendable backward for a prescribed viscosity \(\nu>0\).

Background

Theorem 2.2 establishes that perturbing an analytic initial condition or forcing by almost every scalar multiple of a non-degenerate trigonometric polynomial produces a set of viscosities of Lebesgue measure zero for backward extendability.

Despite this almost-everywhere result, the authors state that their argument does not provide a specific trigonometric polynomial that is known to be non-extendable for an arbitrarily prescribed viscosity. The problem asks for an explicit construction rather than a genericity statement.

References

Neither is it clear how to write down an explicit trigonometric polyomial which is non-extendable for a given $\nu>0$ despite Theorem~\ref{eve2} saying this is almost always the case.

— The Bardos-Tartar Conjecture for the Navier-Stokes Equations  (2610.21530 - Barnes et al., 15 Oct 2026) in Section 2, subsection “Backwards extendability in 3D,” concluding remark after Corollary 2.3