I have this exercise. We put $X = L^2(\Omega)^2 \times L^1_0(\Omega), M = H^1_0(\Omega)^2.$

$$a : X \times X \rightarrow \mathbb{R}; ((\sigma,p),(\tau,\alpha)) \rightarrow (\sigma \otimes \tau)$$

$$b : X \times M \longrightarrow \mathbb{R}; ((\tau,\alpha),v) \longrightarrow (\tau \otimes \nabla v) - (\operatorname{div} v , q)$$

Prouve that the existence of solution to the problem

$$ \begin{cases} &\mbox{find } (\sigma , p) \in X , u \in M;\\ &a((\sigma,p),(\tau,\alpha)) + b((\tau,q),u)=0, \quad\forall (\tau,q) \in X\\ & b((\sigma,p),v) = (f,v) , \quad\forall v \in M \end{cases} $$

implies the existence of solution to the Stokes problem

$$ \begin{cases} & - \Delta w + \nabla \Pi = f\\ & \operatorname{div} w = 0\\ & w=0 \quad\mbox{on } \Gamma \end{cases} $$

and I have no idea how to prove this. Thank's for the help.


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