Assume $\Omega$ is a multi-connected bounded domain in $\mathbb{R}^2$, and function $v\in W^{1,2}_{0}(\Omega)$ satisfies:
$$\text{div} \; v=0$$
Then there exists a stream function $\psi\in W^{2,2}(\Omega)$ such that :
$$\nabla \psi=(-v_2,v_1)$$
How to prove this? I know the method when $v$ is continuous which use the Green Theorem and line integral.
Since $v$ is in $W_0^{1,2}(\Omega)$, extending it by zero outside of the domain gives a function in $W^{2,2}$. Thus, we can replace the multiply-connected domain $\Omega$ (with potentially bad boundary) by a ball $B$.
The existence of $\psi$ such that $\nabla \psi=(-v_2,v_1)$ is a form of the Poincaré lemma about closed form being exact. By definition, $\nabla \psi \in W^{1,2}(B)$. By the Poincaré inequality, this gives $\psi\in L^2(B)$, completing the proof of $\psi\in W^{2,2}(B)$.