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.