Extension criterion involving the middle eigenvalue of the strain tensor on local strong solutions to the 3D Navier–Stokes equations

In this article, we prove an extension criterion for a local strong solution to the 3D Navier–Stokes equations that only require control of the positive part of middle eigenvalue of strain tensor in the critical endpoint Besov space, i.e., ${\lambda }_2^{+}\in L^2(0,T;{\dot{B}}_{\infty ,\infty }^{-1})$. This gives a positive answer to the problem proposed by Miller [1] and improves the results by Wu [2], [3], [4]. The proof relies on the identity for enstrophy growth and $L^p$-norm estimate of the gradient of ${\lambda }_2^{+}$.