Let $\tau$ and $\ell$ be positive numbers. We know that the space $H^2(0,\ell)\cap H^1_0(0,\ell)$ is compactly embedded into $L^6(0,\ell)$. Now, is the space $L^2(0,\tau;H^2(0,\ell)\cap H^1_0(0,\ell))$ compactly embedded into $L^2(0,\tau;L^6(0,\ell))$?
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5$\begingroup$ Well, the space $\mathbb{R}$ embeds compactly into $\mathbb{R}$, but still $L^2(0,\tau;\mathbb{R})$ does not embed compactly into $L^2(0,\tau;\mathbb{R})$, right? $\endgroup$– Jochen GlueckCommented Sep 8, 2018 at 9:10
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