# American Institute of Mathematical Sciences

ISSN:
1556-1801

eISSN:
1556-181X

All Issues

## Networks & Heterogeneous Media

2014 , Volume 9 , Issue 2

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2014, 9(2): 197-216 doi: 10.3934/nhm.2014.9.197 +[Abstract](196) +[PDF](675.3KB)
Abstract:
In this paper we extend some of the previous results for a system of transport equations on a closed network. We consider the Cauchy problem for a flow on a reducible network; that is, a network represented by a diagraph which is not strongly connected. In particular, such a network can contain sources and sinks. We prove well-posedness of the problem with generalized Kirchhoff's conditions, which allow for amplification and/or reduction of the flow at the nodes, on such reducible networks with sources but show that the problem becomes ill-posed if the network has a sink. Furthermore, we extend the existing results on the asymptotic periodicity of the flow to such networks. In particular, in contrast to previous papers, we consider networks with acyclic parts and we prove that such parts of the network become depleted in a finite time, an estimate of which is also provided. Finally, we show how to apply these results to open networks where a portion of the flowing material is allowed to leave the network.
2014, 9(2): 217-238 doi: 10.3934/nhm.2014.9.217 +[Abstract](176) +[PDF](865.6KB)
Abstract:
We analyze Lennard-Jones systems from the standpoint of variational principles beyond the static framework. In a one-dimensional setting such systems have already been shown to be equivalent to energies of Fracture Mechanics. Here we show that this equivalence can also be given in dynamical terms using the notion of minimizing movements.
2014, 9(2): 239-268 doi: 10.3934/nhm.2014.9.239 +[Abstract](239) +[PDF](3338.0KB)
Abstract:
The Aw-Rascle-Zhang (ARZ) model can be interpreted as a generalization of the Lighthill-Whitham-Richards (LWR) model, possessing a family of fundamental diagram curves, each of which represents a class of drivers with a different empty road velocity. A weakness of this approach is that different drivers possess vastly different densities at which traffic flow stagnates. This drawback can be overcome by modifying the pressure relation in the ARZ model, leading to the generalized Aw-Rascle-Zhang (GARZ) model. We present an approach to determine the parameter functions of the GARZ model from fundamental diagram measurement data. The predictive accuracy of the resulting data-fitted GARZ model is compared to other traffic models by means of a three-detector test setup, employing two types of data: vehicle trajectory data, and sensor data. This work also considers the extension of the ARZ and the GARZ models to models with a relaxation term, and conducts an investigation of the optimal relaxation time.
2014, 9(2): 269-297 doi: 10.3934/nhm.2014.9.269 +[Abstract](189) +[PDF](611.3KB)
Abstract:
We analyze the continuous time evolution of a $d$-dimensional system of $N$ self propelled particles with a kinematic constraint on the velocities inspired by the original Vicsek's one [29]. Interactions among particles are specified by a pairwise potential in such a way that the velocity of any given particle is updated to the weighted average velocity of all those particles interacting with it. The weights are given in terms of the interaction rate function. The interaction is not of mean field type and the system is non-Hamiltonian. When the size of the system is fixed, we show the existence of an invariant manifold in the phase space and prove its exponential asymptotic stability. In the kinetic limit we show that the particle density satisfies a nonlinear kinetic equation of Vlasov type, under suitable conditions on the interaction. We study the qualitative behaviour of the solution and we show that the Boltzmann-Vlasov entropy is strictly decreasing in time.
2014, 9(2): 299-314 doi: 10.3934/nhm.2014.9.299 +[Abstract](201) +[PDF](422.5KB)
Abstract:
We consider damped elastodynamic networks where the damping matrix is assumed to be a non-negative linear combination of the stiffness and mass matrices (also known as Rayleigh or proportional damping). We give here a characterization of the frequency response of such networks. We also answer the synthesis question for such networks, i.e., how to construct a Rayleigh damped elastodynamic network with a given frequency response. Our analysis shows that not all damped elastodynamic networks can be realized when the proportionality constants between the damping matrix and the mass and stiffness matrices are fixed.
2014, 9(2): 315-334 doi: 10.3934/nhm.2014.9.315 +[Abstract](207) +[PDF](586.7KB)
Abstract:
In this article, we discuss the optimization of a linearized traffic flow network model based on conservation laws. We present two solution approaches. One relies on the classical Lagrangian formalism (or adjoint calculus), whereas another one uses a discrete mixed-integer framework. We show how both approaches are related to each other. Numerical experiments are accompanied to show the quality of solutions.
2014, 9(2): 335-351 doi: 10.3934/nhm.2014.9.335 +[Abstract](242) +[PDF](390.3KB)
Abstract:
We address the exponential consensus problem for the linearized Vicsek model which was introduced by Jadbabaie et al. in [10] under a joint rooted leadership via the $(sp)$ matrices. This model deals with self-propelled particles moving in the plane with the same speed but different headings interacting with neighboring agents by a linear relaxation rule. When the time-varying switching topology of the neighbor graph satisfies some weak connectivity condition, namely,  joint connectivity condition'' in the spatial-temporal domain, it is well known that the consensus for the linearized Vicsek model can be achieved asymptotically. In this paper, we extend the theory of $(sp)$ matrices and apply it to revisit this asymptotic consensus problem and give an explicit estimate on the maximum Lyapunov exponent, when the underlying network topology satisfies the joint rooted leadership which is directed and non-symmetric.
2014, 9(2): 353-382 doi: 10.3934/nhm.2014.9.353 +[Abstract](425) +[PDF](669.4KB)
Abstract:
We derive a two-scale homogenization limit for reaction-diffusion systems where for some species the diffusion length is of order 1 whereas for the other species the diffusion length is of the order of the periodic microstructure. Thus, in the limit the latter species will display diffusion only on the microscale but not on the macroscale. Because of this missing compactness, the nonlinear coupling through the reaction terms cannot be homogenized but needs to be treated on the two-scale level. In particular, we have to develop new error estimates to derive strong convergence results for passing to the limit.

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