On a two components condition for regularity of the 3D Navier–Stokes equations under physical slip boundary conditions on non-flat boundaries

Abstract

This work concerns the sufficient condition for the regularity of solutions to the evolution Navier–Stokes equations known in the literature as Prodi–Serrin condition. H.-O. Bae and H. J. Choe proved in 1997 that, in the whole space $$\mathbb {R}^3,$$ it is sufficient that two components of the velocity satisfy the above condition in order to guarantee the regularity of solutions. In 2017, H. Beirao da Veiga extended this result (Beirao da Veiga, J Math Anal Appl 453:212–220, 2017) to the half-space case $$\mathbb {R}^n_+$$ under slip boundary conditions by assuming that the velocity components parallel to the boundary enjoy the above condition. It remained open whether the flat boundary geometry is essential. Below, we prove that, under physical slip boundary conditions imposed in cylindrical boundaries, the result still holds.