Well-posedness for scalar conservation laws with moving flux constraints

TL;DR

This paper establishes well-posedness for a coupled ODE-PDE traffic model with moving bottlenecks, proving uniqueness and continuous dependence of solutions using a novel backward in time approach.

Contribution

It introduces a new method to prove well-posedness for scalar conservation laws with moving flux constraints in coupled ODE-PDE systems.

Findings

Proved uniqueness of solutions

Established continuous dependence on initial data

Developed a backward in time analytical method

Abstract

We consider a strongly coupled ODE-PDE system representing moving bottlenecks immersed in vehicular traffic. The PDE consists of a scalar conservation law modeling the traffic flow evolution and the ODE models the trajectory of a slow moving vehicle. The moving bottleneck influences the bulk traffic flow via a point flux constraint, which is given by an inequality on the flux at the slow vehicle position. We prove uniqueness and continuous dependence of solutions with respect to initial data of bounded variation. The proof is based on a new backward in time method established to capture the values of the norm of generalized tangent vectors at every time.