Well-posedness theory for nonlinear scalar conservation laws on networks

Abstract

<p style='text-indent:20px;'>We consider nonlinear scalar conservation laws posed on a network. We define an entropy condition for scalar conservation laws on networks and establish $L^1$ stability, and thus uniqueness, for weak solutions satisfying the entropy condition. We apply standard finite volume methods and show stability and convergence to the unique entropy solution, thus establishing existence of a solution in the process. Both our existence and stability/uniqueness theory is centred around families of stationary states for the equation. In one important case – for monotone fluxes with an upwind difference scheme – we show that the set of (discrete) stationary solutions is indeed sufficiently large to suit our general theory. We demonstrate the method's properties through several numerical experiments.</p>

Burgers’ Equation Revisited: Extension of Mono-Dimensional Case on a Network

“…Condition (23) was also introduced in [12, Def. 5] as one of common assumptions imposed on different transmission solvers considered in the literature.…”

“…The problem of inviscid Burgers equation on networks belongs to the family of conservation laws on networks that has been developed for about thirty years and still receives considerable interest [5,12,23]. The major motivation for studying this topic is traffic modelling, see for instance [8,15,17], initiated with the well-established now Lighthill-Whitham model [20].…”

Well-Posedness and Convergence of a Finite Volume Method for Conservation Laws on Networks

“…In this paper we want to build on the work done in our earlier paper [16]. There, a general framework for the well-posedness of scalar nonlinear conservation laws on graphs was constructed.…”

“…Proposition 2.4 (Musch, Fjordholm, Risebro [16]). Let u k k∈I be a weak solution of (2.1) such that f k • u k (•, t) has a trace at x = 0 for every k ∈ I and a.e.…”

Asymptotic expansion for the solution of a convection-diffusion problem in a thin graph-like junction

“…In various fields, first-order (typically nonlinear) hyperbolic equations on networks like (1.4) have gained interest and led to a wide range of literature in recent years (see, e.g., [2,Chapt. 3], [6,14,19,26] for traffic models and [12] for models of gas flow, and the references therein). As was noted in [14], the dynamics at a vertex is not uniquely determined by imposing the conservation of mass through it and to fully describe the evolution of the process being modeled over the entire graph, the first step is to define the concept of the solution at the vertex.…”