On the extension to slip boundary conditions of a Bae and Choe regularity criterion for the Navier-Stokes equations. The half-space case

TL;DR

This paper extends a regularity criterion for Navier-Stokes solutions, originally proven in the whole space, to the half-space with slip boundary conditions, showing that a single velocity component parallel to the boundary suffices for regularity.

Contribution

It generalizes the Prodi-Serrin regularity criterion to the half-space case with slip boundary conditions, focusing on a single velocity component.

Findings

Regularity criterion extended to half-space with slip boundary conditions.

Sufficiency of a single velocity component parallel to boundary for regularity.

Flat boundary geometry not essential for the criterion.

Abstract

This notes concern the sufficient conditions for regularity of solutions to the evolution Navier-Stokes equations known in the literature as Prodi-Serrin's condition. H.-O. Bae and H.-J. Choe proved in a 1999 paper that, in the whole space R^3, it is merely sufficient that two components of the velocity satisfy the above condition. Below, we extend the result to the half-space case R^n_+ under slip boundary conditions. We show that it is sufficient that the velocity component parallel to the boundary enjoys the above condition. Flat boundary geometry seems not essential, as suggested by some preliminary calculations in cylindrical domains.