Generic critical L3–Besov regularity of inviscid limits

Prove that for generic initial data u0 ∈ L2(𝕋d), inviscid limits of sequences of Leray–Hopf weak solutions of the incompressible Navier–Stokes equations are uniformly bounded in L3(0,T; B3,∞1/3(𝕋d)).

Background

Building on rigorous forms of Kolmogorov’s 4/3 and 4/5 laws proved earlier in the paper, the author argues heuristically that turbulent flows should saturate the one-third Besov regularity in L3, neither exceeding it (which would force vanishing flux) nor falling below it (which would make the flux diverge).

The conjecture proposes a generic uniform L3–Besov 1/3 bound for inviscid limits of Leray–Hopf Navier–Stokes solutions, aligning with model problems (Burgers and Kraichnan) where analogous critical regularization is known, but remaining unproved for Navier–Stokes.