On the regularity of Navier–Stokes equations in critical space

This paper focuses on the regularity of the Navier–Stokes equations in critical space. Let [u(x, t), p(x, t)] denote a suitable weak solution of the Navier–Stokes equations in QT=R3×(−T,0). We prove that if u(x, t) is in the scaling-invariant space Lt∞Lx3p1Lxhp2(QT), where 1p1+2p2=1, 2 < p1 < ∞, and xh = (x1, x2), then u is a smooth solution in QT and does not blow up at t = 0.