Conservation laws on complex networks

Abstract

This paper considers a system described by a conservation law on a general network and deals with solutions to Cauchy problems. The main application is to vehicular traffic, for which we refer to the Lighthill-Whitham-Richards (LWR) model. Assuming to have bounds on the conserved quantity, we are able to prove existence of solutions to Cauchy problems for every initial datum in L 1 loc . Moreover Lipschitz continuous dependence of the solution with respect to initial data is discussed.

Continuous-time link-based kinematic wave model: formulation, solution existence, and well-posedness

“…Notably, Garavello and Piccoli (2009) provide sufficient conditions for the existence of network solutions by specifying a set of abstract properties to be satisfied by the junction models (Riemann Solvers). These sufficient conditions will be illustrated in detail and checked against the two specific junction models considered in this paper.…”

“…Thus there is hardly any universal result on the existence of network solutions. Garavello and Piccoli (2009) are the first to provide sufficient existence conditions by stipulating a few abstract assumptions on the Riemann Solvers considered for the junctions. Our strategy for showing existence is by proving that the two Riemann Solvers presented in Section 2.2 satisfy the sufficient conditions given by Garavello and Piccoli (2009).…”

“…In Garavello and Piccoli (2009), the three sufficient conditions to be satisfied by a Riemann Solver to ensure the existence of a network solution are specified as follows. In rough words such conditions can be expressed as follows:…”

“…Work in this regard includes but is not limited to Bretti et al (2006); Chitour and Piccoli (2005); Coclite et al (2005); D'Apice and Piccoli (2008); Garavello and Piccoli (2009) ;Herty and Klar (2003); Holden and Risebro (1995); Jin (2010); Khoshyaran (1999, 2002) and Ni and Leonard (2005). The Riemann Solver may take various forms, depending on specific assumptions made at the road intersection under consideration.…”

“…The Riemann Solver may take various forms, depending on specific assumptions made at the road intersection under consideration. Some common features of existing Riemann Solvers include the maximization of flow through the junction (Daganzo, 1995;Coclite et al, 2005;Garavello and Piccoli, 2006), the prescribed vehicle turning ratio (Holden and Risebro, 1995;Jin, 2010), and the driving priorities (Coclite et al, 2005;Garavello and Piccoli, 2009).…”