On regularity for the axisymmetric Navier–Stokes equations

In this paper, we establish a regularity criterion for the Navier–Stokes system with axisymmetric initial data. It is proved that if the local axisymmetric smooth solution u   satisfies ‖uθ1r≤ς‖Lα((0,T⁎);Lβ)<∞‖uθ1r≤ς‖Lα((0,T⁎);Lβ)<∞ for any given ς>0ς>0, with 2α+3β=1, 4≤β≤64≤β≤6, then the strong solution keeps smoothness up to time T⁎T⁎.