Abstract:

Two new time-dependent versions of div-curl results in a bounded domain $\domain\subset\RR^3$ are presented. We study a limit of the product $\vector{v}_k\vector{w}_k$, where the sequences $\vector{v}_k$ and $\vector{w}_k$ belong to $\Lp{2}$. In Theorem \ref{lem:kompaktnost} we assume that $\rotor\vector{v}_k$ is bounded in the $L_p$-norm and $ \diver\vector{w}_k$ is controlled in the $L_r$-norm. In Theorem \ref{lem:kompaktnost1} we suppose that $\rotor\vector{w}_k$ is bounded in the $L_p$-norm and $ \diver\vector{w}_k$ is controlled in the $L_r$-norm. The time derivative of $\vector{w}_k$ is bounded in both cases in the norm of $\Hk{-1}$. The convergence (in the sense of distributions) of $\vector{v}_k\vector{w}_k$ to the product $\vector{v}\vector{w}$ of weak limits of $\vector{v}_k$ and $\vector{w}_k$ is shown.