Regularity Criteria for Navier-Stokes Equations with Slip Boundary Conditions on Non-flat Boundaries via Two Velocity Components

TL;DR

This paper extends regularity criteria for the Navier-Stokes equations based on two velocity components to smooth, non-flat boundaries with slip boundary conditions, generalizing previous results on flat and cylindrical domains.

Contribution

It introduces a new regularity criterion applicable to smooth arbitrary boundaries with slip conditions, expanding the scope beyond flat and cylindrical geometries.

Findings

Established regularity criteria for smooth non-flat boundaries.

Demonstrated the applicability of criteria under slip boundary conditions.

Compared new results with existing flat boundary cases.

Abstract

H.-O. Bae and H.J. Choe, in a 1997 paper, established a regularity criteria for the incompressible Navier-Stokes equations in the whole space $\R^3$ based on two velocity components. Recently, one of the present authors extended this result to the half-space case $\R^3_+\,.$ Further, this author in collaboration with J. Bemelmans and J. Brand extended the result to cylindrical domains under physical slip boundary conditions. In this note we obtain a similar result in the case of smooth arbitrary boundaries, but under a distinct, apparently very similar, slip boundary condition. They coincide just on flat portions of the boundary. Otherwise, a reciprocal reduction between the two results looks not obvious, as shown in the last section below.