I'm reading a hand-waving argument in a proof of Chapter 7 of Navier–Stokes Equations by Constantin and Foias. I would like to know if I understand it correctly.
Let $\Omega\subset{\mathbb{R}^n}$ be an open set with $\partial \Omega$ being $C^k$, $k\geq 2$.
Let $\mathcal{V}$ be the space
$$\displaystyle \mathcal{V}=\{u\in C_0^\infty(\Omega)\mid \nabla\cdot u=0\}.$$
Let
$
H=\overline{\mathcal{V}}^{\lVert\cdot\rVert_{L^2(\Omega)^n}}
$
and
$
V=\overline{\mathcal{V}}^{\lVert\cdot\rVert_{H_0^1(\Omega)^n}}.
$
Is the following statement true?
Suppose $u_m\to u$ weakly in $V$. Then there exists a subsequence $u_{m'}\to u$ strongly in $H$.
Best Answer
I think it is correct. since $u_m \to u$ weakly in V, so $u_m$ is bounded ,then (by the compact embedding theorem) $H$ can be embedding $V$,thus $u_{{m}^{\prime}} \to u $ strongly in $H$.