Let $\bf F$ be a smooth vector field in $\mathbb R^3$, which is null outside a finite compact domain $V$. By Helmholtz decomposition thm, there exist a scalar field $\Phi$ and a vector field $\bf A$ such that $${\bf F} = \nabla \Phi + \nabla \times {\bf A}.$$

Do there exist smooth fields $\Phi$ and $\bf A$, which are null outside $V$ and which satisfy this identity?