Let $\bf F$ be a smooth vector field, which is null outside a finite compact domain $V$. By Helmoltz decomposition thm, there exist a scalar field $\Phi$ and a vector field $\bf A$ such that $${\bf F} = \nabla \Phi + \nabla \times {\bf A}.$$
For me, it's seems unlikely that smooth fields $\Phi$ and $\bf A$ which are null outside $V$ cannot be found to satisfy this identity, but I have troubles to prove that. Any idea?
The Helmotz decomposition theorem actually says that, a vector field $F\in L^2(V)$ could be decomposed into $F=A+\nabla\Phi$ where the normal component of vector field $A$ vanishes on $\partial V$ (in the sense of trace) and $\Phi\in H^1(\Omega)$.
Therefore, $F$ vanishes on $\partial V$ must imply that the normal component of $\nabla\Phi$ vanishes on $\partial V$, and it does note necessarily imply that $\Phi$ vanishes on $\partial V$. In fact, if $\Phi$ satisfies the decomposition, then $\Phi+C$ also satisfies.
Reference: section 2.2 of Robinson's book The Three-Dimensional Navier–Stokes Equations.