Recollements associated to cotorsion pairs over upper triangular matrix rings

Let $A$, $B$ be two rings and $T=\left(\begin{smallmatrix} A&M\\0&B\\ \end{smallmatrix}\right)$ with $M$ an $A$-$B$-bimodule. Suppose that we are given two complete hereditary cotorsion pairs $(\mathcal{A}_{A},\mathcal{B}_{A})$ and $(\mathcal{C}_{B},\mathcal{D}_{B})$ in $A$-Mod and $B$-Mod, respectively. We define two cotorsion pairs $(\Phi(\mathcal{A}_{A},\mathcal{C}_{B}),\operatorname{Rep}(\mathcal{B}_{A},\mathcal{D}_{B}))$ and $(\operatorname{Rep}(\mathcal{A}_{A},\mathcal{C}_{B}),\Psi(\mathcal{B}_{A},\mathcal{D}_{B}))$ in $T$-Mod and show that both of these cotorsion pairs are complete and hereditary. If we are given two cofibrantly generated model structures $\mathcal{M}_{A}$ and $\mathcal{M}_{B}$ on $A$-Mod and $B$-Mod, respectively, then using the result above, we investigate when there exists a cofibrantly generated model structure $\mathcal{M}_{T}$ on $T$-Mod and a recollement of $\operatorname{Ho}(\mathcal{M}_{T})$ relative to $\operatorname{Ho}(\mathcal{M}_{A})$ and $\operatorname{Ho}(\mathcal{M}_{B})$. Finally, some applications are given in Gorenstein homological algebra.