When the Zariski space is a Noetherian space

We characterize when the Zariski space $\mathrm{Z}\mathrm{a}\mathrm{r}(K\mathrm{\mid }D)$ (where $D$ is an integral domain, $K$ is a field containing $D$ and $D$ is integrally closed in $K$) and the set $\mathrm{Z}\mathrm{a}{\mathrm{r}}_{\mathrm{m}\mathrm{i}\mathrm{n}}(L\mathrm{\mid }D)$ of its minimal elements are Noetherian spaces.