Muliti-scale regularity of axisymmetric Navier-Stokes equations

Abstract

By applying the delicate \textit{a priori} estimates for the equations of $(\Phi ,\Gamma )$, which is introduced in the previous work, we obtain some multi-scale regularity criteria of the swirl component $u^{\theta }$ for the 3D axisymmetric Navier-Stokes equations. In particularly, the solution $\mathbf{u}$ can be continued beyond the time $T$, provided that $u^{\theta }$ satiesfies $$u^{\theta }\in L_T^p{L}_v^{q_v}{L}_h^{q_h,w},\frac{2}{p}+\frac{1}{q_v}+\frac{2}{q_h}\le 1,2<q_h\le \infty ,\frac{1}{q_v}+\frac{2}{q_h}<1.$$