About weakly convergence in Majda-Bertozzi.

Question

I am studying the proof of the existence of local solutions of the Navier-Stokes equation, in Majda.Bertozzi book, page 109. Fixed $m$, I have a sequence $\{v^{\varepsilon}\}$ such that $v^{\varepsilon}(\cdot,t)\in H^m(\mathbb{R}^3)~\forall t\in[0,T]$. I know that $v^{\varepsilon}\to v$ in the spaces $C([0,T];L^2(\mathbb{R}^3))$, $C([0,T];H^{m'}(\mathbb{R}^3))$ if $m'<m$ and $C([0,T];C^k(\mathbb{R}^3))$ when $k=0,1,2$. Moreover, as $\{v^{\varepsilon}\}$ is bounded in $L^2([0,T];H^m(\mathbb{R}^3))$, then exists a subsequence that converges weakly in this space. How can I prove that the mentioned subsequence tends weakly to $v$. Thanks a lot! Does someone know why $v(\cdot,t)\in H^m(\mathbb{R}^3)$?