# Article

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Keywords:
compensated compactness; convergence; vector fields
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.
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