Div-curl lemma revisited: Applications in electromagnetism

summary:Two new time-dependent versions of div-curl results in a bounded domain $\Omega\subset\mathbb{R}^3$ are presented. We study a limit of the product {\boldmath v}_k{\boldmath w}_k, where the sequences {\boldmath v}_k and {\boldmath w}_k belong to \L_{2}(\Omega). In Theorem 2.1 we assume that \nabla\times{\boldmath v}_k is bounded in the $L_p$-norm and \nabla\cdot{\boldmath w}_k is controlled in the $L_r$-norm. In Theorem 2.2 we suppose that \nabla\times{\boldmath w}_k is bounded in the $L_p$-norm and \nabla\cdot{\boldmath w}_k is controlled in the $L_r$-norm. The time derivative of {\boldmath w}_k is bounded in both cases in the norm of \H^{-1}(\Omega). The convergence (in the sense of distributions) of {\boldmath v}_k{\boldmath w}_k to the product {\boldmath v}{\boldmath w} of weak limits of {\boldmath v}_k and {\boldmath w}_k is shown