Equation 11.23.9313276789DFD

\begin{equation} \begin{split} \frac{\mathrm{d}}{\mathrm{d}t}\int _ X\Phi (\rho _ t)\nu ={}& \frac nM\int _ XG(\rho _ t)\mathrm{dd}^{\mathrm{c}}(\omega _{u_ t}^{n-1})\\ & -\frac nM\int _ X\frac{\Phi ''(\rho _ t)}{\rho _ t} \mathrm{d}\rho _ t\wedge \mathrm d^{\mathrm c}\rho _ t\wedge \omega _{u_ t}^{n-1}. \end{split} \end{equation}