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Networks & Heterogeneous Media

2013 , Volume 8 , Issue 2

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General constrained conservation laws. Application to pedestrian flow modeling
Christophe Chalons, Paola Goatin and Nicolas Seguin
2013, 8(2): 433-463 doi: 10.3934/nhm.2013.8.433 +[Abstract](182) +[PDF](770.2KB)
Abstract:
We extend the results on conservation laws with local flux constraint obtained in [2, 12] to general (non-concave) flux functions and non-classical solutions arising in pedestrian flow modeling [15]. We first provide a well-posedness result based on wave-front tracking approximations and the Kružhkov doubling of variable technique for a general conservation law with constrained flux. This provides a sound basis for dealing with non-classical solutions accounting for panic states in the pedestrian flow model introduced by Colombo and Rosini [15]. In particular, flux constraints are used here to model the presence of doors and obstacles. We propose a "front-tracking" finite volume scheme allowing to sharply capture classical and non-classical discontinuities. Numerical simulations illustrating the Braess paradox are presented as validation of the method.
Weighted energy method and long wave short wave decomposition on the linearized compressible Navier-Stokes equation
Sun-Ho Choi
2013, 8(2): 465-479 doi: 10.3934/nhm.2013.8.465 +[Abstract](157) +[PDF](375.8KB)
Abstract:
The purpose of this paper is to study asymptotic behaviors of the Green function of the linearized compressible Navier-Stokes equation. Liu, T.-P. and Zeng, Y. obtained a point-wise estimate for the Green function of the linearized compressible Navier-Stokes equation in [Comm. Pure Appl. Math. 47, 1053--1082 (1994)] and [Mem. Amer. Math. Soc. 125 (1997), no. 599]. In this paper, we propose a new methodology to investigate point-wise behavior of the Green function of the compressible Navier-Stokes equation. This methodology consists of complex analysis method and weighted energy estimate which was originally proposed by Liu, T.-P. and Yu, S.-H. in [Comm. Pure Appl. Math., 57, 1543--1608 (2004)] for the Boltzmann equation. We will apply this methodology to get a point-wise estimate of the Green function for large $t>0$.
Domain patterns and hysteresis in phase-transforming solids: Analysis and numerical simulations of a sharp interface dissipative model via phase-field approximation
Antonio DeSimone and Martin Kružík
2013, 8(2): 481-499 doi: 10.3934/nhm.2013.8.481 +[Abstract](161) +[PDF](942.7KB)
Abstract:
We propose a sharp-interface model which describes rate-independent hysteresis in phase-transforming solids (such as shape memory alloys) by resolving explicitly domain patterns and their dissipative evolution. We show that the governing Gibbs' energy functional is the $\Gamma$-limit of a family of regularized Gibbs' energies obtained through a phase-field approximation. This leads to the convergence of the solution of the quasistatic evolution problem associated with the regularized energy to the one corresponding to the sharp interface model. Based on this convergence result, we propose a numerical scheme which allows us to simulate mechanical experiments for both spatially homogeneous and heterogeneous samples. We use the latter to assess the role that impurities and defects may have in determining the response exhibited by real samples. In particular, our numerical results indicate that small heterogeneities are essential in order to obtain spatially localized nucleation of a new martensitic variant from a pre-existing one in stress-controlled experiments.
Gamma-expansion for a 1D confined Lennard-Jones model with point defect
Thomas Hudson
2013, 8(2): 501-527 doi: 10.3934/nhm.2013.8.501 +[Abstract](124) +[PDF](499.7KB)
Abstract:
We compute a rigorous asymptotic expansion of the energy of a point defect in a 1D chain of atoms with second neighbour interactions. We propose the Confined Lennard-Jones model for interatomic interactions, where it is assumed that nearest neighbour potentials are globally convex and second neighbour potentials are globally concave. We derive the $\Gamma$-limit for the energy functional as the number of atoms per period tends to infinity and derive an explicit form for the first order term in a $\Gamma$-expansion in terms of an infinite cell problem. We prove exponential decay properties for minimisers of the energy in the infinite cell problem, suggesting that the perturbation to the deformation introduced by the defect is confined to a thin boundary layer.
The stationary behaviour of fluid limits of reversible processes is concentrated on stationary points
Jean-Yves Le Boudec
2013, 8(2): 529-540 doi: 10.3934/nhm.2013.8.529 +[Abstract](119) +[PDF](208.1KB)
Abstract:
Assume that a stochastic process can be approximated, when some scale parameter gets large, by a fluid limit (also called "mean field limit", or "hydrodynamic limit"). A common practice, often called the "fixed point approximation" consists in approximating the stationary behaviour of the stochastic process by the stationary points of the fluid limit. It is known that this may be incorrect in general, as the stationary behaviour of the fluid limit may not be described by its stationary points. We show however that, if the stochastic process is reversible, the fixed point approximation is indeed valid. More precisely, we assume that the stochastic process converges to the fluid limit in distribution (hence in probability) at every fixed point in time. This assumption is very weak and holds for a large family of processes, among which many mean field and other interaction models. We show that the reversibility of the stochastic process implies that any limit point of its stationary distribution is concentrated on stationary points of the fluid limit. If the fluid limit has a unique stationary point, it is an approximation of the stationary distribution of the stochastic process.
Homogenization of hexagonal lattices
Hervé Le Dret and Annie Raoult
2013, 8(2): 541-572 doi: 10.3934/nhm.2013.8.541 +[Abstract](181) +[PDF](766.0KB)
Abstract:
We characterize the macroscopic effective mechanical behavior of a graphene sheet modeled by a hexagonal lattice of elastic bars, using $\Gamma$-convergence.
Asymptotics of an optimal compliance-network problem
Al-hassem Nayam
2013, 8(2): 573-589 doi: 10.3934/nhm.2013.8.573 +[Abstract](109) +[PDF](388.6KB)
Abstract:
We consider the problem of the optimal location of a Dirichlet region in a $d$-dimensional domain $\Omega$ subjected to a given force $f$ in order to minimize the $p$-compliance of the configuration. We look for the optimal region among the class of all closed connected sets of assigned length $l.$ Then we let the length $l$ tend to infinity and we look at the $\Gamma$-limit of a suitable rescaled functional, from which we get information of the asymptotic distribution of the optimal region. We also study the case where the Dirichlet region is a discrete set of finite cardinality.
On the ramified optimal allocation problem
Qinglan Xia and Shaofeng Xu
2013, 8(2): 591-624 doi: 10.3934/nhm.2013.8.591 +[Abstract](188) +[PDF](729.0KB)
Abstract:
This paper proposes an optimal allocation problem with ramified transport technologies in a spatial economy. Ramified transportation is used to model network-like branching structures attributed to the economies of scale in group transportation. A social planner aims at finding an optimal allocation plan and an associated optimal allocation path to minimize the overall cost of transporting commodity from factories to households. This problem differentiates itself from existing ramified transport literature in that the distribution of production among factories is not fixed but endogenously determined as observed in many allocation practices. It is shown that due to the transport economies of scale, each optimal allocation plan corresponds equivalently to an optimal assignment map from households to factories. This optimal assignment map provides a natural partition of both households and allocation paths. We develop methods of marginal transportation analysis and projectional analysis to study the properties of optimal assignment maps. These properties are then related to the search for an optimal assignment map in the context of state matrix.
Errata corrige
Alberto Bressan
2013, 8(2): 625-625 doi: 10.3934/nhm.2013.8.625 +[Abstract](99) +[PDF](200.2KB)
Abstract:
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