Genericity of alignment-induced self-regularization from the 4/5 law

Ascertain whether the limited regularity obtained for incompressible Navier–Stokes turbulence via the Kolmogorov 4/5 law under an alignment hypothesis holds generically, i.e., without imposing special alignment conditions on the flow.

Background

The paper proves strong, viscosity-uniform self-regularization for Burgers turbulence using coercive flux identities. For Navier–Stokes, the only inviscid invariant known to be dissipated is kinetic energy; its 4/5-law flux is not coercive in general.

With an extra alignment hypothesis, prior work shows limited regularity can be inferred from the 4/5 law, but the author explicitly states that whether such self-regularization is a generic property of turbulent Navier–Stokes flows remains open.

References

With an additional alignment hypothesis, the law does confer limited regularity [D22]. Whether this is a generic feature of turbulence is open.

— Mathematical Theorems on Turbulence  (2601.09619 - Drivas, 14 Jan 2026) in Section “Lessons from model problems,” Subsection “Burgers’s equation for pressureless gas,” Remarks following Theorem on self-regularization and intermittency