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 R-3 x (-1, 1) is regular on R-3 x (-1, 0] if partial derivative(3)u belongs to M-2p/(2p-3),M- alpha ((-1, 0); L-p(R-3)) for any alpha > 1 and p is an element of (3/2, infinity), which is a logarithmic-type variation of a Morrey space in time. For each alpha > 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. (C) 2021 Elsevier Ltd. All rights reserved.

Automated summary

The study shows the existence of an appropriate weak solution to the incompressible Navier-Stokes equations under certain conditions, and provides a detailed description of the properties of the solution. The spatial form of the study exhibits a logarithmic-type variation over time and has a critical influence on the scaling properties of the equations.