We introduce a model dealing with conservation laws on networks and coupled boundary conditions at the junctions. In particular, we introduce buffers of fixed arbitrary size and time dependent split ratios at the junctions , which represent how traffic is routed through the network, while guaranteeing spill-back phenomena at nodes. Having defined the dynamics at the level of conservation laws, we lift it up to the Hamilton-Jacobi (H-J) formulation and write boundary datum of incoming and outgoing junctions as functions of the queue sizes and vice-versa. The Hamilton-Jacobi formulation provides the necessary regularity estimates to derive a fixed-point problem in a proper Banach space setting, which is used to prove well-posedness of the model. Finally, we detail how to apply our framework to a non-trivial road network, with several intersections and finite-length links.

中文翻译：

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网络上具有有限缓冲区的宏观交通流模型：利用 Hamilton-Jacobi 方程的适定性

我们引入了一个处理网络守恒定律和连接点耦合边界条件的模型。特别是，我们在连接处引入了固定任意大小和时间相关分流比的缓冲区，这表示流量如何通过网络路由，同时保证节点处的溢出现象。在守恒定律层面定义了动力学，我们将其提升到 Hamilton-Jacobi (HJ) 公式，并将传入和传出连接点的边界数据写入队列大小的函数，反之亦然。Hamilton-Jacobi 公式提供了必要的正则估计，以在适当的 Banach 空间设置中导出不动点问题，用于证明模型的适定性。最后，我们详细介绍了如何将我们的框架应用于非平凡的道路网络，