A fine property of the convective terms of axisymmetric MHD system, and a regularity criterion in terms of $\om^\tt$

In [Zhang, Zujin; Yao, Zheng-an. 3D axisymmetric MHD system with regularity in the swirl component of the vorticity. Comput. Math. Appl. 73 (2017), no. 12, 2573--2580], we have obtained the following fine property of the convective terms of axisymmetric MHD system Let $u,v,w$ be smooth axisymmetric $\bbR^3$-valued functions. Then $$\bee\label{lem:me:equal} \bea &\quad\sum_{i,j,k=1}^3\p_ku_j\cdot \p_jv_i\cdot \p_kw_i\\ &=\frac{u^r}{r}\cdot \frac{v^r}{r}\cdot \frac{w^r}{r} +\frac{u^r}{r}\cdot \frac{v^\tt}{r}\cdot \frac{w^\tt}{r}\\ &\quad+\frac{u^\tt}{r}\cdot \p_rv^r\cdot \frac{w^\tt}{r} -\frac{u^\tt}{r}\cdot \p_rv^\tt\cdot \frac{w^r}{r}\\ &\quad+ \p_ru^\tt\cdot \frac{v^r}{r}\cdot\p_rw^\tt +\p_zu^\tt \cdot \frac{v^r}{r}\cdot \p_zw^\tt -\p_ru^\tt\cdot \frac{v^\tt}{r}\cdot \p_rw^r -\p_zu^\tt\cdot \frac{v^\tt}{r}\cdot \p_zw^r \\ &\quad +\p_ru^r\cdot \p_rv^r\cdot \p_rw^r +\p_ru^r\cdot \p_rv^\tt\cdot \p_rw^\tt +\p_ru^r\cdot \p_rv^z\cdot \p_rw^z\\ &\quad +\p_ru^z\cdot \p_zv^r\cdot \p_rw^r +\p_ru^z\cdot \p_zv^\tt\cdot \p_rw^\tt +\p_ru^z\cdot \p_zv^z\cdot \p_rw^z\\ &\quad +\p_zu^r\cdot \p_rv^r\cdot \p_zw^r +\p_zu^r\cdot \p_rv^\tt\cdot \p_zw^\tt +\p_zu^r\cdot \p_rv^z\cdot \p_zw^z\\ &\quad +\p_zu^z\cdot \p_zv^r\cdot \p_zw^r +\p_zu^z\cdot \p_zv^\tt\cdot \p_zw^\tt +\p_zu^z\cdot \p_zv^z\cdot \p_zw^z. \eea \eee$$ With this above fine property, we could be able to find a regularity criterion in terms of $\om^\tt$ and $j^\tt$. Moreover, using the governing equations of $j^\tt$: $$\bee\label{j_tt} \bea &\p_t j^\tt +u^r\p_rj^\tt+u^z\p_zj^\tt -\sex{\p_r^2+\p_z^2+\frac{1}{r}\p_r-\frac{1}{r^2}}j^\tt\\ &=b^r\p_r\om^\tt +b^z\p_z\om^\tt +(\p_ru^r-\p_zu^z)(\p_zb^r+\p_rb^z) -(\p_zu^r+\p_ru^z) (\p_rb^r-\p_zb^z), \eea \eee$$ we could show that if $$\bee\label{thm:me:om^tt} \om^\tt\in L^p(0,T;L^q(\bbR^3)),\quad\frac{2}{p} +\frac{3}{q}=2,\quad 2\leq q\leq 3, \eee$$ then the solution is smooth on $(0,T)$.