On the blow-up criterion for the Navier–Stokes equations with critical time order.

We prove that every smooth solution u (t , x) on (0 , T) of incompressible Navier–Stokes equations on R n is extensible beyond t > T if u (t , x) ∈ L w r (0 , T ; L σ p) for 2 r + n p = 1 and p > n satisfies blow-up critical time order estimate: ‖ u (t) ‖ L p ≤ ϵ (T − t) − p − n 2 p for T − δ < t < T with sufficiently small positive constants ϵ and δ. Here L w r denote the weak L r space. It is well-known that if the solution u satisfies u ∈ L r (0 , T ; L σ p) with n < p ≤ ∞ and 2 ≤ r < ∞ such that 2 r + n p = 1 then u is extensible continued beyond the time T. In this paper, we consider the blow-up criterion when u ∉ L r (0 , T ; L σ p). [ABSTRACT FROM AUTHOR]