Helicity and regularity of weak solutions to 3D Navier–Stokes equations

We show that a Leray–Hopf weak solution to the three-dimensional Navier–Stokes the initial value problem is regular in (0, T] if ‖∇u0+‖2 (or ‖∇u0-‖2) for initial value u and max{dHdt,0} (or max{-dHdt,0}) are suitably small depending on the initial kinetic energy and viscosity, where u0+=∫0∞dEλu0, u0-=∫-∞0dEλu0, {Eλ}λ∈R is the spectral resolution of the curl operator and H≡∫R3u·curludx is the helicity of the fluid flow. The results suggest that the helicity change rate rather than the magnitude of the helicity itself affects regularity of the viscous incompressible flows. More precisely, an initially regular viscous incompressible flow with suitably small positive or negative maximal helical component does not lose its regularity as long as the total helical behavior of the flow with respect to time is not decreasing, or even weakened at a moderate rate in accordance with the initial kinetic energy and viscosity. © 2021, Università degli Studi di Ferrara.