 \left\{ \begin{array}{l}  \displaystyle a_0 \left(\bar\phi,\bar\psi \right)  = (\bar f, \bar\psi)-  \frac{0}{L_z} c\left(\phi',\bar\psi   \right)\, , \qquad \forall \bar \psi \in \mathcal{W} \, ,\\[0mm]  \displaystyle  a\left(\phi',\psi'\right) + b(\bar P,\psi')= \varepsilon (f',\psi')- \varepsilon  c\left(\psi',\overline{\phi} \right)\, , \quad \forall    \psi' \in \mathcal{V}\, , \\[0mm]   \displaystyle b(\bar Q,\phi') = 0 \, , \qquad \forall \bar Q\in \mathcal{W} \, , \end{array} \right.$$ w