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Consider a smooth open Set $\Omega\subset\mathbb{R}^3$ and a bounded sequence of vector fields $(u_n)_n \in L^2(\Omega)$ having $0$ divergence. I know how to extend this sequence to the whole space into a divergence-free sequence bounded in $L^2(\mathbb{R}^3)$. My question is : if furthermore $(\text{curl}\hspace{1mm} u_n)_n$ is converging in $L^2(\Omega)$, is it possible to build an extension (still divergence-free, and still bounded in $L^2(\mathbb{R}^3)$) such as the corresponding curls converge in $L^2(\mathbb{R}^3)$ ?

Thanks in advance !


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