Let $\Omega\subseteq \mathbb{R}^{3}$ be a contractible region and $f:\Omega\to \mathbb{R}^{3}$ be a vector field on $f$. If $f$ is smooth vector field and $\nabla\cdot f=0$, then $f=\nabla\times g$ for some smooth vector field $g:\Omega\to \mathbb{R}^{3}$ by de Rham's theorem. ($\because H_{dR}^{2}(\Omega)=H^{2}(\Omega)=0$)
But recently, I heard that this result does not holds for non-smooth vector fields in general. In 2007, theorem 4.1 of this paper shows that if $\Omega$ is contractible region in $\mathbb{R}^{3}$ and $f$ is vector field on $\Omega$ with $\nabla\cdot f=0$ (here $f$ is not necessarily smooth). Then if $f$ is $L^{1}$, there exists (unique) $g\in L^{3/2}(\Omega, \mathbb{R}^{3})$ s.t. $f=\nabla\times g$.
Actually, I have 2 questions :
1) Is there any non-smooth vector field on simply-connected domain s.t. $\nabla\cdot f=0$ but $f=\nabla\times g$ does not have a solution?
2) Is there any cohomology theory that considers non-smooth vector fields? (for example, $C^{k}$ vector fields for some $k\geq 0$?)
Thanks!