Muliti-scale regularity of axisymmetric Navier-Stokes equations

Abstract

By applying the delicate \textit{a priori} estimates for the equations of $(Φ,Γ)$, which is introduced in the previous work, we obtain some multi-scale regularity criteria of the swirl component $u^θ$ for the 3D axisymmetric Navier-Stokes equations. In particularly, the solution $\mathbf{u}$ can be continued beyond the time $T$, provided that $u^θ$ satiesfies $$ u^θ \in L^{p}_{T}L^{q_{v}}_{v}L^{q_{h},w}_{h},~~\frac{2}{p}+\frac{1}{q_{v}}+\frac{2}{q_{h}}\leq 1, ~2<q_{h}\leq\infty,~\frac{1}{q_{v}}+\frac{2}{q_{h}}<1. $$