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

Abstract

We show that a Leray–Hopf weak solution to the three-dimensional Navier–Stokes the initial value problem is regular in (0, T] if $\Vert \nabla u_0^+\Vert _2$ (or $\Vert \nabla u_0^-\Vert _2$) for initial value $u_0$ and $\max \{\frac{d{\mathcal H}}{dt},0\}$ (or $\max \{-\frac{d{\mathcal H}}{dt},0\}$) are suitably small depending on the initial kinetic energy and viscosity, where $u_0^+=\int _0^{\infty } dE_\lambda u_0$, $u_0^-=\int _{-\infty }^0 dE_\lambda u_0$, $\{E_\lambda \}_{\lambda \in {\mathbb R}}$ is the spectral resolution of the $\mathrm{curl}$ operator and ${\mathcal H}\equiv \int _{\mathbb R^3}u\cdot \mathrm{curl} u\,dx$ 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.