Viscous profiles in models of collective movements with negative diffusivities

TL;DR

This paper studies traveling-wave solutions in one-dimensional advection-diffusion equations with negative diffusivity, relevant for modeling aggregation in traffic and crowd dynamics, establishing existence, uniqueness, and behavior of such solutions.

Contribution

It proves existence, uniqueness, and sharpness of viscous profiles connecting states with different diffusivity signs, extending to cases with sign-changing diffusivity and real-world data.

Findings

Existence and uniqueness of traveling-wave solutions with negative diffusivity.

Profiles are sharp and avoid plateaus under certain conditions.

Conditions are satisfied by a large class of real data diffusivities.

Abstract

In this paper we consider an advection-diffusion equation, in one space dimension, whose diffusivity can be negative. Such equations arise in particular in the modeling of vehicular traffic flows or crowds dynamics, where a negative diffusivity simulates aggregation phenomena. We focus on traveling-wave solutions that connect two states whose diffusivity has different signs; under some geometric conditions we prove the existence, uniqueness (in a suitable class of solutions avoiding plateaus) and sharpness of the corresponding profiles. Such results are then extended to the case of end states where the diffusivity is positive but it becomes negative in some interval between them. Also the vanishing-viscosity limit is considered. At last, we provide and discuss several examples of diffusivities that change sign and show that our conditions are satisfied for a large class of them in correspondence of real data.