Nonlinear degenerate evolution equations in mixed formulation. (English) Zbl 1237.47079
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.
Reviewer: Yongxiang Li (Lanzhou)
MSC:
47J35 | Nonlinear evolution equations |
47H05 | Monotone operators and generalizations |
35F25 | Initial value problems for nonlinear first-order PDEs |
35Q35 | PDEs in connection with fluid mechanics |
76S05 | Flows in porous media; filtration; seepage |