$$ex cdot{f b}=0 a imes [( imes {f b}) imes {f b}]= imes [ cdot ({f b}otimes {f b})]. eex$$
证明: 右端第一个分量为 $$eex ea &quad sum_i p_2(p_i(b_ib_3))-p_3(p_i(b_ib_2))\ &=sum_i p_2(b_ip_ib_3)-p_3(b_ip_ib_2)\ &=sum_i p_2b_ip_ib_3-p_3b_ip_ib_2\ &quad +sum_ib_ip_i(p_2b_3-p_3b_2)\ &=p_2b_1p_1b_3+p_2b_2p_2b_3+p_2b_3p_3b_3\ &quad-p_3b_1p_1b_2-p_3b_2p_2b_2-p_3b_3p_3b_2\ &quad+({f b}cdot )j_1\ &=p_2b_1p_1b_3-p_2b_3p_1b_1\ &quad -p_3b_1p_1b_2+p_3b_2p_1b_1\ &quad+({f b}cdot )j_1\ &=p_2b_1p_1b_3 -p_3b_1p_1b_2 -j_1p_1b_1 +({f b}cdot )j_1\ &=-(p_3b_1-p_1b_3)p_2b_1 -(p_1b_2-p_2b_1)p_3b_1 -j_1p_1b_1+({f b}cdot )j_1\ &=-j_2p_2b_1-j_3p_3b_1-j_1p_1b_1 +({f b}cdot )j_1\ &=-({f j}cdot )b_1+({f b}cdot )j_1. eea eeex$$ 利用公式 (link) $$ex imes({f a} imes{f b})=({f b}cdot ){f a} -({f a}cdot ){f b}+{f a}( cdot{f b})-{f b}( cdot{f a}), eex$$ 我们知 $$ex ({f j} imes {f b})=({f b}cdot ){f j}-({f j}cdot ){f b}. eex$$ 而左端的第一项也为 $-({f j}cdot )b_1+({f b}cdot )j_1$. 故有结论.
see [D. Chae, M. Schonbek, On the temporal decay for the Hall-magnetohydrodynamic equations, J. Differential Equations, 255 (2013), 3971--3982].