Introduction
Let $Q \subset \mathbb{R}^3$ a bounded connected regular domain.
Lets denote $A$ and $B$ the following spaces :
$A$ the space of $3 \times 3$ function matrix with coefficient valued in $L^2(Q)$, noted $G$, such as there exists $u \in (H^1_0(Q))^3, \ div \ u =0$ in $Q$ such as $G=\nabla u$ (i.e $G$ derives from a divergence free potential).
$B$ the space of $3 \times 3$ function matrix with coefficient valued in $L^2(Q)$, noted $F$, that verifies
$$ \int_Q F:\nabla \varphi \ \mathrm{d} x=0, \quad \forall \varphi \in (H^1_0(Q))^3, \quad div \ \varphi =0 \text{ in } Q$$
One can verify easily that the following orthogonal decomposition for the space of $3 \times 3$ function matrix with coefficient valued in $L^2(Q)$ :
$$\mathcal{M}_3(L^2(Q))= A \ \oplus^\perp B$$ with the scalar product $<F|G>=\int_Q F:G \ \mathrm{d}x$ with the symbol $":"$ being the usual scalar product for matrices : $C:D=\sum_{1 \leq i,j \leq 3} c_{ij} d_{ij}$.
($A$ is indeed closed in $\mathcal{M}_3(L^2(Q))$ using Poincaré inequality...)
My question
Let's take $F_n$ and $G_n$ two sequences in $A$, respectivly $B$. We assume that
-$F_n$ weakly converges toward $G_0$ in $\mathcal{M}_3(L^2(Q))$ (i.e $\int_Q G_n :u \underset{n \rightarrow +\infty}{\longrightarrow} \int_Q G_0:u$ for any $u$ in $\mathcal{M}_3(L^2(Q))$)
-$G_n$ weakly converges toward $G_0$ in $\mathcal{M}_3(L^2(Q))$
Now, for a fixed function $\varphi \in C^\infty_0(Q))$, I am looking at the following quantity :
$$\int_Q \varphi F_n: G_n \ \mathrm{d} x$$ and I would like to show that it converges toward $$\int_Q \varphi F_0: G_0\ \mathrm{d} x .$$
This look look exactly like a div/curl formula with a twist on the hypothesis through the divergence free potential. I have tried to prove it the same way using :
$$\int_Q \varphi F_n : \nabla u_n \ \mathrm{d} x = \int_Q F_n : \nabla (u_n \varphi) \ \mathrm{d} x - \int_Q F_n : (u_n \nabla \varphi^t) \ \mathrm{d} x$$ where $G_n=\nabla u_n$.
The last term converges thanks to Rellich injection that gives me that $u_n$ converges strongly in $(L^2(Q))^3$, and therefore I have a scalar product between a weak converging term and a strong converging term :
$$\int_Q F_n : (u_n \nabla \varphi^t) \ \mathrm{d} x \underset{n \rightarrow + \infty}{\longrightarrow} \int_Q F_0 : (u_0 \nabla \varphi^t) \ \mathrm{d} x.$$
However, for the first integral, I can't conclude anything since $div(u \varphi)$ is not $0$, and so I can't use that $F_n$ belongs to $B$...
Any ideas are welcomed.
I was thinking of maybe using the Leray Projector $P$ that projects a function of $L^2(Q)$ on the space of divergence free $L^2(Q)$ functions., but then I need to be able to control the following quantity :
$$\int_Q F_n : \nabla (P (u_n \varphi) - u_n \varphi) \ \mathrm{d} x$$