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

TL;DR

This paper establishes a new anisotropic regularity criterion for the 3D incompressible Navier-Stokes equations, showing regularity under a logarithmic-type Morrey space condition on the vertical derivative of the velocity.

Contribution

It introduces a novel anisotropic regularity condition involving a logarithmic Morrey space, extending the class of initial data ensuring regularity for all exponents.

Findings

Regularity achieved under the new anisotropic condition.

The space is critical with respect to the Navier-Stokes scaling.

Includes all subcritical $L^q_tL^p_x$ spaces for the vertical derivative.

Abstract

We show that a suitable weak solution to the incompressible Navier-Stokes equations on ${\mathbb{R}^3\times(-1,1)}$ is regular on $\mathbb{R}^3\times(0,1]$ if $\partial_3 u $ belongs to $M^{2p/(2p-3),\alpha } ((-1,0);L^p (\mathbb{R}^3 ))$ for any $\alpha >1$ and $p\in (3/2,\infty)$, 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 $L^q ((-1,0);L^p (\mathbb{R}^3 ))$ that are subcritical, that is for which $2/q+3/p<2$.