On the interaction of strain and vorticity for solutions of the Navier--Stokes equation

Abstract

In this paper, we prove a new identity for divergence free vector fields, showing that \begin{equation*} \left<-ΔS,ω\otimesω\right>=0, \end{equation*} where $S_{ij}=\frac{1}{2}\left({\partial }_i{u}_j+{\partial }_j{u}_i\right)$ is the symmetric part of the velocity gradient, and $\omega =\nabla \times u$ is the vorticity. This identity will allow us to understand the interaction of different aspects of the nonlinearity in the Navier--Stokes equation from the strain and vorticity perspective, particularly as they relate to the depletion of the nonlinearity by advection. We will prove global regularity for the strain-vorticity interaction model equation, a model equation for studying the impact of the vorticity on the evolution of strain which has the same identity for enstrophy growth as the full Navier--Stokes equation. We will also use this identity to obtain several new regularity criteria for the Navier--Stokes equation, one of which will help to clarify the circumstances in which advection can work to deplete the nonlinearity, preventing finite-time blowup.

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