Let $V \subset H$ be a dense and compact embedding. Let $$\lVert u\_n\rVert\_{L^\infty(0,T;H)} + \lVert u\_n \rVert\_{L^2(0,T;V)} < C$$ where $C$ is independent of $n$. It follows that eg. $u\_n \rightharpoonup u$ in $L^2(0,T;V)$ and $u\_n \rightharpoonup^\* u$ in $L^\infty(0,T;H)$ for some $u$.



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> **Does this imply $u\_n \to u$ in $L^2(0,T;H)$ strongly?**
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I have seen this claim on page 11 of <http://www.mat.unimi.it/users/rocca/cgquad2.ps> (see equation 3.41) but I find it hard to believe. The reference cited there is a book by Lions in which I cannot find anything.


I would like a reference if thiis is true. Thanks.



$u_n$ bounded in $L^\infty(0,T;H) \cap L^2(0,T;V)$ implies $u_n \to u$ strongly in $L^2(0,T;H)$?
As it is stated, this property does not hold: indeed consider the sequence of functinos $(u\_n)\_{n \in \mathbb{N}}$ defined for $t \in [0, T]$ by 
$$
 u\_n (t) = \sin (2n\pi t) v,
$$
where $v \in V$ is a fixed vector. 
The sequence converges clearly weakly to $0$ in $L^2 (0, T; V)$. 
Also since $L^2 (0, T)$ is dense in $L^1 (0, T)$ and the squence $(\sin 2n\pi t)$ is bounded in $L^\infty (0, T)$, the sequence $(\sin 2n\pi t)\_{n \in \mathbb{N}}$ converges weakly-\* in $L^\infty (0, T)$, and thus the sequence $(u\_n)\_{n \in \mathbb{N}}$ converges weakly-\* in $L^\infty (0, T, H)$.
This sequence satisfies thus the assumptions, but not the conclusion.


Classical results on this topic involve an assumption of the type $(u\_n')\_{n \in\mathbb{N}}$ is bounded in $L^2 ([0, T], V')$ and imply the strong convergence (see for example Evans, *Partial differential equations*, 1998, section 5.9.2).


