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

TL;DR

This paper introduces a new mathematical identity relating strain and vorticity in Navier-Stokes solutions, enabling better understanding of nonlinearity interactions and establishing global regularity results and criteria for the equations.

Contribution

It proves a novel identity for divergence-free vector fields, facilitating analysis of nonlinearity depletion and regularity in Navier-Stokes equations.

Findings

Established a new identity for divergence-free vector fields involving strain and vorticity.

Proved global regularity for a strain-vorticity interaction model equation.

Derived new regularity criteria clarifying conditions for nonlinearity depletion.

Abstract

In this paper, we prove a new identity for divergence free vector fields, showing that \begin{equation*} \left<-\Delta S,\omega\otimes\omega\right>=0, \end{equation*} where $S_{ij}=\frac{1}{2}\left(\partial_iu_j+\partial_ju_i\right)$ 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.