In this paper, we show that a strong solution u to the Navier–Stokes equations on the time interval (0, T) can be continued beyond time T if either one row of the strain tensor satisfies \(S_{3j} \in L^{\frac{2}{2 - r}}\left( 0, T; \dot{V}_{\infty , \infty , \theta }^{-r}\right) \) for some \(r \in (0, 1)\) , \(\theta \in [1, \infty ]\) , and \(j = 1, 2, 3\) , or if the positive part of the middle eigenvalue of the strain tensor satisfies \(\lambda _2^{+} = \max \{ \lambda _2(x), 0 \} \in L^{\frac{2}{2 - r}}\left( 0, T; \dot{V}_{\infty , \infty , \theta }^{-r} \right) \) for \(r \in (0, 1)\) and \(\theta \in [1, \infty ]\) . Here, \(\dot{V}_{p, q, \theta }^{s}\) is a Banach space that may be larger than the homogeneous Besov space \(\dot{B}_{p, q}^{s}\) . Our approach relies on a generalized Vishik-type trilinear estimate, which is of independent interest.
