A Regularity Criterion for the Navier–Stokes Equation Involving Only the Middle Eigenvalue of the Strain Tensor

Abstract

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 4920.75—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.