if \begin{equation} \left\{ \begin{array}{l} (u^{0\nu },u^{1\nu },v^{0\nu },v^{1\nu },p^{0\nu },q^{0\nu }) \rightarrow (u^{0},u^{1},v^{0},v^{1},p^{0},q^{0}) \\ \text{strongly in } (H^1_\gamma \cap H^2(0,L)\times H^1_\gamma \times (H^1_\gamma \cap H^2(0,L))\times H^1_\gamma \times H^1(0,L)\times H^1_\gamma). \end{array} \right. \end{equation}
where $L>0$, $H^1_\gamma = \{ \Phi \in H^1(0,L) \ \Phi (0)=0 \}$ and $ (u^{0\nu },u^{1\nu },v^{0\nu },v^{1\nu },p^{0\nu },q^{0\nu })$ is the initial data of the problem.
We fix $\nu \in \mathbb{N}$. If K=$\{ u^{0\nu },u^{1\nu },v^{0\nu } ,v^{1\nu },p^{0\nu },q^{0\nu }\}$ is a linearly independent set.
How costruct a basis depending of $K$ for write the solution in the following form
$$u^{\nu m}(t)=\sum_{j=1}^{m}\alpha^{j\nu m}(t)w_j^{\nu}(x),\hspace{1.4cm}v^{\nu m}(t)=\sum_{j=1}^{m}\beta^{j\nu m}(t)w_j^{\nu}(x),$$ $$p^{\nu m}(t)=\sum_{j=1}^{m}\gamma^{j\nu m}(t)w_j^{\nu}(x),\hspace{0.3cm}\text{and}\hspace{0.3cm} q^{\nu m}(t)=\sum_{j=1}^{m}\sigma^{j\nu m}(t)w_j^{\nu}(x).$$