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

Abstract

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., λ2+∈L2(0,T;Ḃ∞,∞−1). This gives a positive answer to the problem proposed by Miller [1] and improves the results by Wu [2–4]. The proof relies on the identity for enstrophy growth and Lp-norm estimate of the gradient of λ2+.