TITLE: A question on the proof of the Serrin condition for the regularity of Navier-Stokes equations
QUESTION [5 upvotes]: Edit: This question has been substantially modified on January 12th, 2015.

I have been studying Michael Struwe's paper "On Partial Results for the Navier-Stokes Equations", Comm. Pure Appl. Math 41 (1988), no. 4, 437-458, and I have a question regarding the rigorous justification of the formal calculations in the proof of the Serrin condition in Section 3 of the paper. I have also studied the original proof by Serrin.
Consider the weak solutions of the Navier-Stokes system. Let's assume $n=3$ for simplicity. A weak solution $u \in L^{2, \infty}(Q; \mathbb R^3)$, with $|\nabla u| \in L^2(Q)$, is defined by requiring that $\nabla \cdot u=0$ and 
$$
\int_Q -u \cdot \partial_t \varphi + \nabla u \cdot \nabla \varphi - [(u \cdot \nabla)\varphi \cdot u] \, dx \, dt =0
$$
for every $\varphi \in C_0^\infty(Q)$ with $\nabla \cdot \varphi=0$.
By choosing a test function $\varphi = \nabla \times \psi$ with a compactly supported smooth $\psi=(\psi_1, \psi_2, \psi_3)$ it is possible to show that the vorticity $\omega = \text{curl}\  u$ satisfies the vorticity equation
$$\tag 2
-\sum_{i=1}^3 \int_Q \omega_i\partial_t\psi_i+\omega_i\Delta \psi_i+ \sum_{j=1}^3 (u_i\omega_j-\omega_iu_j)\frac{\partial \psi_i}{\partial x_j} \, dx \, dt= 0.
$$
Then, in the proof by Struwe, one formally chooses a test function $\psi=\omega|\omega|^{2s-2}\zeta^2$, where $\zeta$ is a standard cut-off function. In order to rigorously justify this formal choice of the test function, one considers an approximated vorticity equation (2) with $u_i$ replaced by $u_i^k$, where $u_i^k$ is any smooth approximation of $u_i$ in $L^2$-norm. Then one obtains an approximative solution $\omega_i^k$, which has enough regularity for the formal calculations to be rigorously justified. 
The final step of the proof is to show that $\omega_i^k \to \omega_i$ weakly in $L^2$. By some analysis, it is possible to show that $\omega_i^k$, indeed, converge to a function $\tilde \omega_i$, and also that for each $k$, the function $\omega_i^k$ is a unique solution of the approximative equation.
Now my question is that how do we know that $\omega_i = \tilde\omega_i$? 
I believe that it is possible to choose the approximative sequence $u_i^k$ to be smooth solutions of the Navier-Stokes equation. In this case, it is clear that $\omega_i^k = \nabla \times u_i^k$ and, as discussed below in the comments section, then we may use the continuity of differential operators in the distributional topology to conclude that $\omega_i^k = \nabla \times u_i^k \to \nabla \times u = \omega_i$ as $k \to \infty$. However, it seems like a non-trivial result that $u_i^k$ can be chosen to be smooth solutions of the Navier-Stokes equation, and Struwe does not state or prove such result. 
I am mostly interested in the methodology rather than the result itself and, thus, I would like to understand if it is possible to avoid using the fact that $u_i^k$ solves the equation. In particular, if $u_i^k$ is any smooth approximation of $u_i$, I do not see why $\omega_i^k$ needs to satisfy $\omega_i^k = \nabla \times u_i^k$. Perhaps this is something which follows from the fact that $\omega_i^k$ is a unique solution of the approximative equation? My first guess would be to try showing by Biot-Savart law that $\nabla \times u_i^k$ is always a solution and then to rely on uniqueness, but I would prefer not to reinvent the wheel if someone knows how to do this.

REPLY [2 votes]: If you read one more line down from equation (3.1), Struwe helpfully points you to 

(see Serrin [15], (11), p.190)

which happens to be this paper and indeed, around page 190 you find the claim proved in detail. Please note that the notion of weak solution in this particular part of the manuscript may be slightly different from the sense you included in the statement of you questions; see also the first few pages of Serrin's article for what he meant by weak solutions.