Let us consider $\mathscr D(\mathbb R^3;\mathbb R^3)$ the space of smooth compactly supported vector fields in $\mathbb R^3$ and let us define
$\widetilde{\mathscr D}=\mathscr D(\mathbb R^3;\mathbb R^3)\cap \ker\text{div}$
the space of smooth compactly supported vector fields in $\mathbb R^3$ with null divergence.

Question 1: is it true that
$
\widetilde{\mathscr D}=\text{curl }
\mathscr D(\mathbb R^3;\mathbb R^3)
$?

Question 2: is it true that the topological dual ${(\widetilde{\mathscr D})}{'}$ of $\widetilde{\mathscr D}$ satisfies
$$
 {(\widetilde{\mathscr D})}{'}=\text{curl }
\mathscr D'(\mathbb R^3;\mathbb R^3)=\mathscr D'(\mathbb R^3;\mathbb R^3)\cap \ker\text{div}?
$$
These questions are implicitly important to define weak solutions of Euler or Navier-Stokes equations.