# Prove the equivalence of the solution existence of Stokes equation's variational problem to the original problem

I have this exercise. We put $X = L^2(\Omega)^4 \times L^1_0(\Omega), M = H^1_0(\Omega)^2.$

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

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

Prouve that the existence of solution to the problem

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

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.

• who can help me please – jijiii May 26 '13 at 19:17
• What is $\otimes$? – Shuhao Cao May 28 '13 at 22:47
• And probably you copied your variational formulation wrong, you might wanna check the letters. – Shuhao Cao May 28 '13 at 22:53
• hi: he is the product tensoriel. – jijiii May 31 '13 at 16:56
• Then in $\sigma \otimes \tau$, $\sigma$ and $\tau$ are both matrices function? It does not look like this in your definition of spaces. In your definition $\sigma$ and $\tau$ are both $L^2$-integrable vector fields. – Shuhao Cao May 31 '13 at 17:01