\begin{align}
    \partial_t \rho^{(r)} + \left(\nu_r\Gamma_r+\nu_l\Gamma_l\right)\cdot\rho^{(r)} &= \nu_r\Gamma_r\rho^{\Sigma};\\
    \partial_t \rho^{(l)} + \left(\nu_r\Gamma_r+\nu_l\Gamma_l\right)\cdot\rho^{(l)} &= \nu_l\Gamma_l\rho^{\Sigma},
\label{eq:decoupled_sys}
\end{align}
