Nonlinear Diﬀer. Equ. Appl. (2017) 24:55
2017 Springer International Publishing AG
published online August 22, 2017
Nonlinear Diﬀerential Equations
and Applications NoDEA
Particle approximation of a constrained
model for traﬃc ﬂow
To Prof. Alberto Bressan.
Florent Berthelin and Paola Goatin
Abstract. We rigorously prove the convergence of the micro–macro limit
for particle approximations of the constrained pressureless gas dynamics
system. The lack of BV bounds on the density variable is supplied by a
compensated compactness argument.
Mathematics Subject Classiﬁcation. Primary 35L87; Secondary 35L75,
Keywords. Conservation laws with density constraint, Many-particle sys-
tem, Traﬃc ﬂow models.
Macroscopic traﬃc ﬂow models usually consist of partial diﬀerential equations
describing the evolution of aggregated quantities, like traﬃc density and mean
velocity. They express the mass conservation and eventually the traﬃc acceler-
ation. In this article, we focus on a pressure-less gas dynamics system subject
to a maximal density constraint: the constrained pressureless gas dynamics
(CPGD) model, which was introduced in  and can be derived through a sin-
gular limit in the pressure term of a modiﬁed Aw–Rascle–Zhang model [2,14].
Indeed, we start from the Aw–Rascle–Zhang (ARZ) model,
ρ + ∂
(ρ(v + p(ρ))) + ∂
(ρv(v + p(ρ))) = 0,
which is a very well accepted model for traﬃc ﬂow. We observe that, in this
model, upper bounds on the density are not necessarily preserved through the
time evolution of the solution. In practice, the density of cars is bounded from
above by a maximal density ρ
corresponding to a bumper to bumper situation.
This article is part of the topical collection “Hyperbolic PDEs, Fluids, Transport and Appli-
cations: Dedicated to Alberto Bressan for his 60th birthday” guest edited by Fabio Ancona,
Stefano Bianchini, Pierangelo Marcati, Andrea Marson.