# American Institute of Mathematical Sciences

June  2015, 10(2): 255-293. doi: 10.3934/nhm.2015.10.255

## Conservation law models for traffic flow on a network of roads

 1 Penn State University Mathematics Dept., University Park, State College, PA 16802 2 Department of Mathematics, Penn State University, University Park, PA 16802, United States

Received  June 2014 Revised  January 2015 Published  April 2015

The paper develops a model of traffic flow near an intersection, where drivers seeking to enter a congested road wait in a buffer of limited capacity. Initial data comprise the vehicle density on each road, together with the percentage of drivers approaching the intersection who wish to turn into each of the outgoing roads.
If the queue sizes within the buffer are known, then the initial-boundary value problems become decoupled and can be independently solved along each incoming road. Three variational problems are introduced, related to different kind of boundary conditions. From the value functions, one recovers the traffic density along each incoming or outgoing road by a Lax type formula.
Conversely, if these value functions are known, then the queue sizes can be determined by balancing the boundary fluxes of all incoming and outgoing roads. In this way one obtains a contractive transformation, whose fixed point yields the unique solution of the Cauchy problem for traffic flow in an neighborhood of the intersection.
The present model accounts for backward propagation of queues along roads leading to a crowded intersection, it achieves well-posedness for general $L^\infty$ data, and continuity w.r.t. weak convergence of the initial densities.
Citation: Alberto Bressan, Khai T. Nguyen. Conservation law models for traffic flow on a network of roads. Networks & Heterogeneous Media, 2015, 10 (2) : 255-293. doi: 10.3934/nhm.2015.10.255
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