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 ‖ u θ 1 r ≤ ς ‖ L α ( ( 0 , T ⁎ ) ; L β ) < ∞ for any given ς > 0 , with 2 α + 3 β = 1 , 4 ≤ β ≤ 6 , then the strong solution keeps smoothness up to time T ⁎ . [ABSTRACT FROM AUTHOR]