The author considers the system of evolution equations in mixed formulation \[ \begin{matrix} \frac{d}{dt}\,\mathcal{E}_1 u(t)+ \mathcal{A} u(t)+\mathcal{B}p(t)=f(t)\text{ in }V',\\ \frac{d}{dt}\,\mathcal{E}_2 p(t)- \mathcal{B} u(t)+\mathcal{C}p(t)=g(t)\text{ in }Q',\\ u(t)\in V,\;p(t)\in Q ,\end{matrix} \] where \(V\) and \(Q\) are Hilbert spaces and the operators \(\mathcal{E}_1\), \(\mathcal{A}\) map \(V\) to \(V'\), \(\mathcal{E}_2\), \(\mathcal{C}\) map \(Q\) to \(Q'\), and \(\mathcal{B}\) maps \(V\) to \(Q'\). \(\mathcal{E}_1\) and \(\mathcal{E}_2\) are linear and symmetric operators permitted to be degenerate. \(\mathcal{A}\) and \(\mathcal{C}\) are permitted to be nonlinear, but they are maximal monotone. Some existence results on the degenerate Cauchy problem are obtained. The results are illustrated by a Darcy-Stokes coupled system with multiple nonlinearities.