We study evolution equations on metric graphs with reservoirs, that is graphs where a one-dimensional interval is associated to each edge and, in addition, the vertices are able to store and exchange mass with these intervals. Focusing on the case where the dynamics are driven by an entropy functional defined both on the metric edges and vertices, we provide a rigorous understanding of such systems of coupled ordinary and partial differential equations as (generalized) gradient flows in continuity equation format. Approximating the edges by a sequence of vertices, which yields a fully discrete system, we are able to establish existence of solutions in this formalism. Furthermore, we study several scaling limits using the recently developed framework of EDP convergence with embeddings to rigorously show convergence to gradient flows on reduced metric and combinatorial graphs. Finally, numerical studies confirm our theoretical findings and provide additional insights into the dynamics under rescaling.

翻译：我们研究度量图上带储层的演化方程，这类图中每条边关联一个一维区间，且顶点能够存储质量并与这些区间交换质量。聚焦于动力学由定义在度量边和顶点上的熵泛函驱动的情形，我们为此类耦合常微分与偏微分方程系统提供了（广义）连续性方程形式梯度流的严格数学理解。通过用顶点序列逼近边，从而得到完全离散系统，我们能够在该形式体系中证明解的存在性。此外，我们利用近期发展的嵌入式EDP收敛框架研究多种尺度极限，严格证明了向简化度量图与组合图上梯度流的收敛性。数值研究最终验证了理论结果，并为尺度变换下的动力学提供了新的见解。