Let $(M, g)$ be a Riemanian manifold (or $\mathbb{R}^n$ if you prefer). A TT-tensor is a symmetric 2-tensor $\sigma_{ab}$ satisfying
$g^{ab} \sigma_{ab} \equiv 0$ ($\sigma$ is trace free),
$\nabla^a \sigma_{ab} = 0$ ($\sigma$ is divergence free).
These tensors appear in general relativity to construct initial data. I have seen that these tensors also occur in fluid dynamics (any reference would be welcome!). It is known that there exist TT-tensors with compact support at least on some particular $(M, g)$ (see e.g. https://arxiv.org/abs/1003.0535).
I suspect that TT-tensors with compact support are dense (for some topology) in the space of TT-tensors.
Has such a claim been proved for Euclidean space (or any other kind of space)?
One potential proof of this fact is by using the Bach tensor (I will not indicate function spaces in what follows). Indeed, if $(M, b)$ is Einstein and if $g$ is close to $b$ then the divergence of the Bach tensor is quadratic in $g-b$ thus the image of the linearized Bach tensor $\mathcal{B}$ (at $b$) consists of TT-tensors. If one can prove that the image is dense then, given any TT-tensor $\sigma$ and any $\epsilon > 0$, one can find a 2-tensor $h$ such that $\|\sigma - \mathcal{B}(h)\|\leq \epsilon/2$. Then, by choosing an appropriate cutoff function $\chi$, one can arrange that $\|\mathcal{B}(h) - \mathcal{B}(\chi h)\|\leq \epsilon/2$ thus $\|\sigma - \mathcal{B}(\chi h)\|\leq \epsilon$. As $\mathcal{B}(\chi h)$ has compact support, this would prove the claim.