Preservation of AD via forcings

We show that assuming $\mathsf{Z}\mathsf{F}+{\mathsf{A}\mathsf{D}}^{+}+$ “$V=\mathrm{L}(\mathrm{\wp }(\mathbb{R}))$”, any poset which increases $\mathrm{\Theta }$ does not preserve the truth of $\mathsf{A}\mathsf{D}$. We also show that in $\mathsf{Z}\mathsf{F}+\mathsf{A}\mathsf{D}$, any non-trivial poset on $\mathbb{R}$ does not preserve the truth of $\mathsf{A}\mathsf{D}$. This answers the question of Chan and Jackson. Furthermore, we show that under the assumptions $\mathsf{Z}\mathsf{F}+{\mathsf{A}\mathsf{D}}^{+}+$ “$V=\mathrm{L}(\mathrm{\wp }(\mathbb{R}))$” + “$\mathrm{\Theta }\text{ is regular}$”, there is a poset on $\mathrm{\Theta }$ which adds a new subset of $\mathrm{\Theta }$ while preserving the truth of $\mathsf{A}\mathsf{D}$. This answers the question of Cunningham.