在 [Zhang, Zujin. Regularity criteria for the three dimensional Ericksen–Leslie system in homogeneous Besov spaces. Comput. Math. Appl. 75 (2018), no. 3, 1060--1065] 中, 我们讨论了 $$\bee\label{EL:Simple} \seddm{ \p_t\bbu +(\bbu\cdot\n)\bbu -\lap\bbu+\n P =-\n\cdot[\n\bbd \odot\n\bbd],\\ \p_t\bbd+(\bbu\cdot\n)\bbd =\lap \bbd -\bbf(\bbd),\\ \Div\bbu=0,\\ (\bbu,\bbd)|_{t=0}=(\bbu_0,\bbd_0), } \eee$$ 说明如果 $$\bee\label{thm:EL:Simple:reg} \bbu\in L^\frac{2}{1+r}(0,T;\dot B^r_{\infty,\infty}(\bbR^3)),\quad 0<r<1, \eee$$ 则解光滑. 也讨论了 $$\bee\label{EL:d=1} \seddm{ \p_t\bbu +(\bbu\cdot\n)\bbu -\lap \bbu +\n P=-\n\cdot (\n\bbd\odot\n\bbd),\\ \p_t\bbd+(\bbu\cdot\n)\bbd =\lap\bbd+|\n\bbd|^2\bbd,\\ \Div\bbu=0,\quad |\bbd|=1,\\ (\bbu,\bbd_0)|_{t=0}=(\bbu_0,\bbd_0). } \eee$$ 说明如果 $$\bee\label{thm:EL:Simple:d=1:reg} \bbu\in L^\frac{2}{1+r}(0,T;\dot B^r_{\infty,\infty}(\bbR^3)),\quad \n\bbd\in L^\frac{2}{1+s}(0,T;\dot B^s_{\infty,\infty}(\bbR^3)),\quad -1<r,s<1, \eee$$ 则解光滑. 最后讨论了一般的 Ericksen-Leslie 系统 $$\bee\label{EL} \seddm{ \p_t\bbu +(\bbu\cdot\n)\bbu -\lap\bbu +\n P =-\Div \sez{(\n \bbd)^t \cfrac{\p W(\bbd,\n\bbd)}{\p (\n\bbd)}},\\ \p_t\bbd +(\bbu\cdot\n)\bbd =\bbh-(\bbd\cdot \bbh)\bbd,\\ \Div\bbu=0,\quad |\bbd|=1,\\ (\bbu,\bbd)|_{t=0}=(\bbu_0,\bbd_0), } \eee$$ 说明如果 $$\bee\label{thm:EL:reg} \bbu\in L^\frac{2}{1+r}(0,T;\dot B^r_{\infty,\infty}(\bbR^3)),\quad \n\bbd\in L^\frac{2}{1+s}(0,T;\dot B^s_{\infty,\infty}(\bbR^3)),\quad -1<r,s<1, \eee$$ 则解光滑.
链接: https://pan.baidu.com/s/1raiKJeO 密码: eqfb