  This manuscript derives an evolution equation for the symmetric part of the
gradient of the velocity (the strain tensor) in the incompressible
Navier-Stokes equation on $\mathbb{R}^3$, and proves the existence of $L^2$
mild solutions to this equation. We use this equation to obtain a simplified
identity for the growth of enstrophy for mild solutions that depends only on
the strain tensor, not on the nonlocal interaction of the strain tensor with
the vorticity. The resulting identity allows us to prove a new family of
scale-critical, necessary and sufficient conditions for the blow-up of a
solution at some finite time $T_{max}<+\infty$, which depend only on the
history of the positive part of the second eigenvalue of the strain matrix.
Since this matrix is trace-free, this severely restricts the geometry of any
finite-time blow-up. This regularity criterion provides analytic evidence of
the numerically observed tendency of the vorticity to align with the
eigenvector corresponding to the middle eigenvalue of the strain matrix. This
regularity criterion also allows us to prove as a corollary a new scale
critical, one component type, regularity criterion for a range of exponents for
which there were previously no known critical, one component type regularity
criteria. Furthermore, our analysis permits us to extend the known time of
existence of smooth solutions with fixed initial enstrophy
$E_0=\frac{1}{2}\left\|\nabla \otimes u^0\right \|_{L^2}^2$ by a factor of
$4{,}920{.}75$---although the previous constant in the literature was not
expected to be close to optimal, so this improvement is less drastic than it
sounds, especially compared with numerical results. Finally, we will prove the
existence and stability of blow-up for a toy model ODE for the strain equation.
