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

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

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

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

EDIT: this question can be reformulated in the following way: Assume that $\bf F$ is a smooth vector field with compact support. Does there exist a scalar smooth solution $f$ to $$\nabla^2 f = \nabla \cdot \bf F$$ whose support is included inside the support of $\bf F$?

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

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

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