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 R 3 × ( − 1 , 1 ) is regular on R 3 × ( 0 , 1 ] if ∂ 3 u belongs to M 2 p ∕ ( 2 p − 3 ) , α ( ( − 1 , 0 ) ; L p ( R 3 ) ) 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 L q ( ( − 1 , 0 ) ; L p ( R 3 ) ) that are subcritical, that is for which 2 ∕ q + 3 ∕ p 2 .