Div-curl problems and stream functions in 3D Lipschitz domains

We consider the problem of recovering the divergence-free velocity field $\mathbf{U}\in\mathbf{L}^2(\Omega)$ of a given vorticity $\mathbf{F}=\mathrm{curl}\,\mathbf{U}$ on a bounded Lipschitz domain $\Omega\subset \mathbb{R}^3$. To that end, we solve the 'div-curl problem' for a given $\mathbf{F}\in\bigl[ \mathbf{H}_0(\mathrm{curl};\Omega)\bigr]'$. The solution is given in terms of a vector potential (or stream function) $\mathbf{A}\in\mathbf{H}^1(\Omega)$ such that $\mathbf{U}=\mathrm{curl}\,{\mathbf{A}}$. After discussing existence and uniqueness of solutions and associated vector potentials, we propose a well-posed construction for the stream function. A numerical example of the construction is presented at the end.