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

Abstract

We prove that every smooth solution u(t,x) on (0,T) of incompressible Navier–Stokes equations on Rn is extensible beyond t>T if u(t,x)∈Lwr(0,T;Lσp) for 2r+np=1 and p>n satisfies blow-up critical time order estimate: ‖u(t)‖Lp≤ϵ(T−t)−p−n2p for T−δ<t<T with sufficiently small positive constants ϵ and δ. Here Lwr denote the weak Lr space. It is well-known that if the solution u satisfies u∈Lr(0,T;Lσp) with n<p≤∞ and 2≤r<∞ such that 2r+np=1 then u is extensible continued beyond the time T. In this paper, we consider the blow-up criterion when u∉Lr(0,T;Lσp).