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### Open Access Journals

KRM

We continue research on generalized macroscopic models of
conservation type as started in [15]. In this paper we keep the
characteristic (for traffic) non-locality removed in [15] by
Taylor expansion and discuss the merits and problems of such an
expansion. We observe that the models satisfy maximum principles and
conclude that "triggers'' are needed in order to cause traffic jams
(braking waves) in traffic guided by such models. Several such
triggers are introduced and discussed. The models are refined
further in order to properly address non-monotonic (in speed)
traffic regimes, and the inclusion of an individual reaction time is
discussed in the context of a braking wave. A number of numerical
experiments are conducted to exhibit our findings.

KRM

From a Vlasov-type kinetic equation with nonlocal braking and
acceleration forces, taken as a traffic model for higher densities,
we derive macroscopic equations generalizing the second order model
of conservation laws suggested by Aw and Rascle [1] and Zhang
[19]. The nonlocality remains present in these equations,
but more conventional, local equations are derived by using suitable
Taylor expansion. A second order model of this type is discussed in
some detail and is shown to possess traveling wave solutions that
resemble stop-and-go waves in dense traffic. A phase space analysis
suggests that inside the class of such traveling waves
there are
steady solutions that are stable.

KRM

We present an extended discussion of a macroscopic traffic flow
model [18] which includes non-local and relaxation terms
for vehicular traffic flow on unidirectional roads.
The braking and acceleration forces are based on a behavioural model and on
free flow dynamics. The latter are modelled by using different fundamental diagrams.
Numerical investigations for different situations illustrate the properties of the mathematical model. In particular, the emergence of stop-and-go waves is observed for suitable parameter ranges.

KRM

The Cucker-Smale model for flocking or swarming of birds or insects is generalized to scenarios
where a typical bird will be subject to a) a friction force term driving it to fly at optimal speed,
b) a repulsive short range force to avoid collisions, c) an attractive "flocking" force computed
from the birds seen by each bird inside its vision cone, and d) a "boundary" force which will
entice birds to search for and return to the flock if they find themselves at some distance from the
flock. We introduce these forces in detail, discuss the required cutoffs and their implications and
show that there are natural bounds in velocity space. Well-posedness of the initial value problem
is discussed in spaces of measure-valued functions. We conclude with a series of numerical simulations.

KRM

We obtain new a priori estimates for spatially inhomogeneous solutions of a kinetic equation for granular media, as first proposed in [3] and, more recently, studied in [1]. In particular, we show that a family of convex functionals on the phase space is non-increasing along the flow of such equations, and we deduce consequences on the asymptotic behaviour of solutions. Furthermore, using an additional assumption on the interaction kernel and a ``potential for interaction'', we prove a global entropy estimate in the one-dimensional case.

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