Abstract

We prove a new identity for divergence free vector fields, showing that

where $S_{ij}=\frac{1}{2}({\partial }_i{u}_j+{\partial }_j{u}_i)$ is the symmetric part of the velocity gradient, and $\omega =\nabla <!--FUNCTION\; APPLICATION-->\times u$ is the vorticity. This identity allows 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 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 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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