The following definitions/theorems/etc come from this paper, by Masuda, 1984.
Statement of the Navier-Stokes Problem
For $n \geq 3$, let $\Omega \subseteq \mathbb{R}^n$, $T > 0$. We write the incompressible NS equations as follows:
(NS)$\begin{cases} \text{d}_t u - \Delta u + (u \cdot \nabla) u + \nabla p = f; \ \nabla \cdot u = 0, \ x \in \Omega, \ 0<t<T \\ u|_{\partial \Omega} = 0 ; \ u|_{t=0} = a \end{cases}$
Where $u : \Omega \times [0,T) \rightarrow \mathbb{R}^n$ is the unknown vector velocity,
$ p : \Omega \times [0,T) \rightarrow \mathbb{R}^n$ is the unknown pressure,
$ f : \Omega \times [0,T) \rightarrow \mathbb{R}^n $ is a given external force, and
$a:\Omega \rightarrow \mathbb{R}^n$ is the given initial velocity.
Function Spaces
\begin{align} C^{\infty}_{0, \sigma}(\Omega) & := \{ f \in C^{\infty}_{0}(\Omega)^{n} \ | \ \nabla \cdot f = 0 \} \\ L_{\sigma}^{2}(\Omega) & := \overline{C^{\infty}_{0, \sigma}(\Omega)}^{||\cdot||_{L^2}} \\ H^1(\Omega) = W^{1,2}(\Omega) & := \{ f \in L^2 \ | \ \partial_{x_i} f \in L^2, \ i=1,...,n \} \\ H^{1}_{0,\sigma}(\Omega) & := \overline{C^{\infty}_{0, \sigma}(\Omega)}^{||\cdot||_{H^1}} \\ Y &:= H^{1}_{0,\sigma}(\Omega) \cap L^n \end{align}
Assumptions on $a$ and $f$
(1.) $a \in L^2_\sigma$
(2.) $f(\cdot , t) \in L^2$ for almost all $t \in (0,T)$, and $Pf \in L^1(0,t ; L^2_\sigma)$, where $P$ is the Helmholtz projection from $L^2$ into $L^2_\sigma$.
Definition of Weak Solution
A weak solution of (NS) is a function, $u$, on $\Omega \times [0,T)$ such that:
(1.) $u \in L^2(0,T' ; H^{1}_{0,\sigma})$, for all $T' \in (0,T)$
(2.) $u \in L^{\infty}(0,T ; L^{2}_{\sigma})$
(3.) $\int^s_{r} -(u,\partial_{t}\Phi) + (\nabla u , \nabla \Phi) + (u \cdot \nabla u, \Phi) \text{d}t = \int^s_r (f, \Phi) \text{d}t - (u(s), \Phi(s)) + (u(r), \Phi(r))$ for all $0 \leq r \leq s < T$ and all $\Phi \in H^1(r,s; Y)$.
Theorem (Masuda, 1984)
There exists a weak solution, $u$, of (NS). In particular, this function is weak continuous in $L^2_\sigma$ with respect to $t \in [0,T)$, and
$||u(t)||^2_{L^2(\Omega)} + 2 \int^{s}_{0} ||\nabla u(t)||^{2}_{L^2(\Omega)} \text{d}t \leq 2 \int^{s}_{0} (f, u) \text{d}t + ||a||^{2}_{L^2(\Omega)}$ for all $0 \leq t < T$ (Energy Inequality)
and $\lim_{t \rightarrow 0}||u(t) - a||_{L^2(\Omega)} = 0 \ \ \ $ (Continuity at $t=0$)
Here, $(u,f)$ denotes the inner-product of $u$ and $f$ in $L^2(\Omega)$. That is, $(u,f) = \int_{\Omega} u \cdot f \ \text{d}x$.
We explain the meaning of weak continuity of $u$. $u$ is weakly continuous in $L^2(\Omega)$ with respect to $t \in [0,T)$ if:
$(u(t), g) = \int_{\Omega} u(t) \cdot g \ \text{d}x$ is continuous in $t$, for all $g \in L^2_\sigma$.
Proof of Existence (Outline)
(Please feel free to skip this and see the remaining problem of continuity at $t=0.$)
I have proven the existence of a weak solution, $u$, as described in the definition above. I have also shown that this $u$ is weakly continuous in $L^2_\sigma$ with respect to $t$ and satisfies the Energy Inequality.
We prove this using a Galerkin Method. We find a countable set $\{\phi_l \}_{l=1}^{\infty} \subseteq C^{\infty}_{0,\sigma}$ whose span is dense in $Y$ and in $L^{2}_{\sigma}$ (with respect to their respective norms).
We then define $u_{m}(x,t) = \sum^m_{l=1} c_{ml}(t) \phi_{l}(x)$, where $c_{ml}$ are solutions in $C^1(0,T)$ to the below locally Lipschitz ODE:
\begin{cases}\text{d}_t c_{ml} + \sum^m_{i=1} (\nabla \phi_{i} , \nabla \phi_{l})c_{mi} + \sum^m_{i,j=1} (\phi_i \nabla \phi_{j} , \nabla \phi_{l})c_{mi}c_{mj} = (f,\phi_l), \ \text{for all } l=1,...,m \\ c_{ml}(0) = (a, \phi_l), \ \text{for all } l=1,...,m. \end{cases}
We find that these $u_m$ are uniform bounded in $L^2(0,T' ; H^{1}_{0,\sigma})$, for all $T' \in (0,T)$ and in $L^{\infty}(0,T ; L^{2}_{\sigma})$. Thus, since these are Hilbert spaces, there exists subsequences which converge weakly. It is easy to show that $u_m$ convergese to the same function, $u$, in both spaces. Thus we have satisfied (1.) and (2.) from the definition of Weak Solutions. We can use Arzela-Ascoli to show that this $u$ is weak continuous, as described above. See this previous question for details on the weak convergence and weak continuity of $u$.
Using the weak convergence and definition of $u_m$, we show that this $u$ satisfies the Energy Inequality. Using weak convergence and several Lemmas from Masuda's paper, we show that $u$ satisfies the integral equation (3.) from the definition of Weak Solutions.
Last Remaining Problem, Continuity in $L^2$ at $t=0$
It remains only to show that $\lim_{t \rightarrow 0}||u(t) - a||_{L^2(\Omega)} = 0$. In fact, Masuda makes no mention of this property in his proof of the theorem!!!
We know that $u$ is weakly continuous with respect to $t$, as explained above. Also, by the definition of $u_m$, each $u_m$ is continuous with respect to $t$. That is, $u_m \in C((0,T); L^2_{\sigma})$, for all $m$.
I suspect this should be enough to obtain the final result, but I am struggling to show it without requiring uniform continuity. Here is my initial attempt to prove:
$\lim_{t \rightarrow 0}||u(t) - a||_{L^2(\Omega)} = \lim_{t \rightarrow 0}(u(t) - a, u(t) - a) $
$ = \lim_{t \rightarrow 0} \lim_{m \rightarrow \infty}(u(t) - a, u_{m}(t)) - (u_{m}(t) - a, a)$
By an earlier step in the existence proof, we know that $(u_{m}(t), g)$ converges as $m \rightarrow \infty$ uniformly with respect to $t$. (Again, see this previous question for details.) Thus the term on the right, $(u_{m}(t) - a, a)$, converges uniformly with respect to $t$. Therefore, we are able to swap the position of the limits in $t$ and $m$. By definition of $u_m$, it is clear that $\lim_{m \rightarrow \infty} ||u_m(t=0)||_{L^2(\Omega)} = ||a||_{L^2(\Omega)}$, so the term on the right vanishes.
We are stuck with the term on the left, $\lim_{t \rightarrow 0} \lim_{m \rightarrow \infty}(u(t) - a, u_{m}(t))$. As demonstrated in another previous question, we know that, for each fixed $m$, this is continuous in $t$. So we would be able to show that this term vanishes, if we are able to swap the limits in $t$ and $m$.
I know we could do this if we have uniform convergence again, but I am not sure if this is possible, as we now have dependence on $t$ for both sides of the inner-product. We do have uniform boundedness of this term, as the $u_m$ are uniformly bounded in $L^2(0,T' ; H^{1}_{0,\sigma})$, for all $T' \in (0,T)$ and $L^{\infty}(0,T ; L^{2}_{\sigma})$, so we could obtain uniform convergence by showing equicontinuity. But I'm not sure if this can be done.
In short, the question is as follows: can we swap the limits in $t$ and $m$ in $\lim_{t \rightarrow 0} \lim_{m \rightarrow \infty}(u(t) - a, u_{m}(t))$? Or is there another way to show that $\lim_{t \rightarrow 0}||u(t) - a||_{L^2(\Omega)} = 0$?
Consider a sequence $t_n\to 0^+$. If you use the energy inequality you get $$\limsup_n ||u(t_n)||^2_{L^2(\Omega)} \leq ||a||^{2}_{L^2(\Omega)}.$$ So now you should prove that $u(t_n)\ \rightharpoonup a$ in $L^2(\Omega)$. This implies that $$||a||^{2}_{L^2(\Omega)}\le \liminf_n ||u(t_n)||^2_{L^2(\Omega)}$$by the lower semicontinuity of the norm with respect to weak convergence. Hence, you have $$||a||^{2}_{L^2(\Omega)}=\lim_n ||u(t_n)||^2_{L^2(\Omega)}.$$ This implies that $u(t_n)\ \to a$ in $L^2(\Omega)$ (see weak implies strong ).
To prove that $u(t_n)\ \rightharpoonup a$ in $L^2(\Omega)$, use weak continuity of $u$ at $t=0$ to conclude that $u(t_n)\ \rightharpoonup u(0)$ in $L^2(\Omega)$ and the fact that for every $f\in L^2_\sigma(\Omega)$ from the previous question $(f,u_{m_k}(0)\,)\to (f,u(0)\,)$. On the other hand, $$u_{m_k}(\cdot,0) = \sum^{m_k}_{l=1} (a, \phi_l)\phi_{l}\to a \text{ in }L^2(\Omega).$$By uniqueness of limits, this implies that $u(0)=a$.