An anisotropic regularity condition for the 3D incompressible Navier–Stokes equations for the entire exponent range

We show that a suitable weak solution to the incompressible Navier–Stokes equations on R3×(?1,1) is regular R3×(?1,0] if ?3u belongs M2p/(2p?3),?((?1,0);Lp(R3)) for any ?>1 and p?(3/2,?), which logarithmic-type variation of Morrey space in time. For each this is, up logarithm, critical with respect scaling equations, contains all spaces Lq((?1,0);Lp(R3)) are subcritical, 2/q+3/p<2.