Div-curl lemma revisited: Applications in electromagnetism

Slodička, Marián ; Buša, Ján Jr.

Kybernetika, Tome 46 (2010) no. 2, pp. 328-340

Résumé

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.

Détail

Classification : 35B05, 65J10, 65M99, 78A25
Keywords: compensated compactness; convergence; vector fields