\onslide<1-> Applying the decomposition $\vec{u}\approx\vec{u}_{meas}+\vec{w}$ into the original equation and writing a variational problem for $\vec w$ we have the following: Find $(\vec w(t) ,p(t))\in H^1_0(\Omega)\times L^2(\Omega)$ such that
\onslide<1-> Applying the decomposition $\vec{u}=\vec{u}_{meas}+\vec{w}$ into the original equation and writing a variational problem for $\vec w$ we have the following:\\
Find $(\vec w(t) ,p(t))\in H^1_0(\Omega)\times L^2(\Omega)$ such that
There exists a unique solution of Problem \ref{eq:Corrector_discrete} under condition: $$\rho/\tau+ C_\Omega^{-2}\mu/2-\rho3\|\nabla\vec u_{meas}^k\|_\infty > 0$$ for all $k>0$.
There exists a unique solution of Problem (\ref{eq:Corrector_discrete}) under the condition: $$\rho/\tau+ C_\Omega^{-2}\mu/2-\rho3\|\nabla\vec u_{meas}^k\|_\infty > 0$$ for all $k>0$.
\end{theorem}
\onslide<2->
We can furthermore prove the following energy balance:
\onslide<3->
\begin{theorem} For $(\vec w^k ,p^k)$ solution of Problem \ref{eq:Corrector_discrete}, with $\ell_j(\vec v,q)=0$ it holds
\begin{theorem} For $(\vec w^k ,p^k)$ solution of Problem (\ref{eq:Corrector_discrete}), with $\ell_j(\vec v,q)=0$ it holds