In the proof of the existence of weak solutions to the NSE (*[Navier-Stokes Equations](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=Foias&s5=Navier-Stokes%20Equations&s6=&s7=&s8=Books&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=2&mx-pid=972259)* by Constantin and Foias, Chapter 8), the following argument is made:



> 
> Let $u\_m$ converges weakly to $u$ in $L^2(0,T;V)$ where 
>  $$
> V=\overline{\{f\in (C\_0^\infty(\Omega))^n\mid \nabla\cdot f=0\}}^{H^1(\Omega)}
> $$ By extracting a subsequence, we may assume that $u\_m(t\_0)$ converges to $u(t\_0)$ weakly in $V$ for all $t\_0\in[0,T]\setminus E$ for some $E$ of Lebesgue measure $0$. 
> 
> 
> 




---


I don't understand why this is true. The statement would not be true when $V=\mathbb{R}$ according to the comments to a previous [question](https://mathoverflow.net/q/229898/14319). Would anyone point me to the related theorems in known references (or sketch the proof)?



In the proof of the existence of weak solutions to the NSE
*To provide some context*, first let $\Omega$ be a bounded domain in $\mathbb{R}^3$ with smooth boundary.


The popular way of constructing weak solutions is by Galerkin approximation with the eigenfunctions of the Stokes operator (due to Hopf in 1951, or see Ladyzhenskaya's [book](http://rads.stackoverflow.com/amzn/click/1614276714) Chapter 6, Section 6). A second way is to solve the mollified equations
$$ u\_t + \text{div} (u \otimes u\_\rho) - \Delta u + \nabla p = f, $$
where $u\_\rho$ is the mollification of $u$ with an appropriate test function. This is what Leray did for the unbounded case, but it is possible in a bounded domain as well.


In either case, you obtain a sequence of functions $u^{(N)}$ which is bounded in $L^\infty\_t L^2\_x(\Omega \times (0,T))$ and $L^2\_t H^1\_{x,0}(\Omega \times (0,T))$. By boundedness, there is a subsequence $u^{(N)}$ converging in an appropriate sense (weak-$\ast$ in the former case, weak in the latter).


Others have remarked that the above modes of convergence are not enough to obtain a subsequence converging pointwise a.e. (indeed, for the target space $\mathbb{R}$, weak convergence in $L^2(0,T)$ not does imply pointwise convergence a.e.).


*To obtain pointwise a.e. convergence in time*, we can prove strong convergence in a space like $L^2\_t L^2\_x(\Omega \times (0,T))$. To do this, consider the [Aubin-Lions lemma](https://en.wikipedia.org/wiki/Aubin%E2%80%93Lions_lemma) with $p=2$ and the triple
$$ H\_0^1 \overset{\text{cpt}}\hookrightarrow L^2 \hookrightarrow H^{-3}.$$
The compactness of the first embedding follows from the boundedness of $\Omega$ and the Rellich-Kondrachov theorem. Aubin-Lions tells us that
$$ W = \{ f \in L^2\_tH\_0^1(\Omega \times (0,T)) : f\_t \in L^2\_tH\_x^{-3}(\Omega \times (0,T)) \} \overset{\text{cpt}}\hookrightarrow L^2\_tL^2\_x(\Omega \times (0,T)).$$
I choose to bound the derivative in $H\_x^{-3}(\Omega)$ because I want the use of the Sobolev embedding theorem in $n=3$. The nice thing about Aubin-Lions is that you can take as negative a Sobolev space as you want.


To meet the requirements of Aubin-Lions, we remark that the sequence $u^{(N)}$ is already bounded in $L^2\_tH^1\_{x,0}$. It remains to bound the derivative in $L^2\_tH\_x^{-3}$. Something like the following should be attainable from the equation:
$$ || \partial\_t u^{(N)} ||\_{L^2\_t H\_x^{-3}} \lesssim ||u^{(N)}||\_{L^\infty\_tL^2\_x}^2 + ||u^{(N)}||\_{L^2\_t H^1\_x} + ||f^{(N)}||\_{L^2\_t H^{-1}\_x} \lesssim 1. $$
Aubin-Lions then tells us you can choose a subsequence such that
\begin{align}
u^{(N)} \to u &\text{ in } L^2\_tL^2\_x(\Omega \times (0,T)) \\
u^{(N)} \overset{\ast}\rightharpoonup u &\text{ in } L^\infty\_tL^2\_x(\Omega \times (0,T)) \\
u^{(N)} \rightharpoonup u &\text{ in } L^2\_tH^1\_x(\Omega \times (0,T)).
\end{align}
The strong convergence gives you pointwise a.e. convergence on the time interval, which is what you asked for.


Please note: I have essentially given you a poor rewording of the argument in [Seregin's lecture notes](http://rads.stackoverflow.com/amzn/click/9814623407) (see Chapter 5), and I am mostly just rerecording it here. There is a preprint of his notes which comes up on Google, if you want.


For the case $\Omega = \mathbb{R}^3$, the Aubin-Lions lemma about compactness is not available (at least, not globally). In such case, you can look at what Leray did in his original 1934 paper, which is available [here](http://www.math.cornell.edu/~bterrell/leray.pdf) in English and French.


