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

Abstract

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