# Models for dense multilane vehicular traffic

## TL;DR

This paper develops a mathematical model for dense, multi-lane vehicular traffic, analyzing its well-posedness, bounds, and the limit as the number of lanes becomes large, resulting in a continuum traffic flow model.

## Contribution

It introduces a multi-lane traffic model with lane-changing dynamics, proves its well-posedness, and derives a continuum limit as the number of lanes increases.

## Key findings

- The multi-lane model is well-posed with specific bounds.
- Lane-changing behavior is proportional to velocity differences.
- The model converges to a continuum limit as lanes increase.

## Abstract

We study vehicular traffic on a road with multiple lanes and dense, unidirectional traffic following the traditional Lighthill-Whitham-Richards model where the velocity in each lane depends only on the density in the same lane. The model assumes that the tendency of drivers to change to a neighboring lane is proportional to the difference in velocity between the lanes. The model allows for an arbitrary number of lanes, each with its distinct velocity function. The resulting model is a well-posed weakly coupled system of hyperbolic conservation laws with a Lipschitz continuous source. We show several relevant bounds for solutions of this model that are not valid for general weakly coupled systems. Furthermore, by taking an appropriately scaled limit as the number of lanes increases, we derive a model describing a continuum of lanes, and show that the $N$-lane model converges to a weak solution of the continuum model.