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Networks & Heterogeneous Media

2013 , Volume 8 , Issue 4

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Asymptotic periodicity of flows in time-depending networks
Fatih Bayazit , Britta Dorn and  Marjeta Kramar Fijavž
2013, 8(4): 843-855 doi: 10.3934/nhm.2013.8.843 +[Abstract](41) +[PDF](935.3KB)
We consider a linear transport equation on the edges of a network with time-varying coefficients. Using methods for non-autonomous abstract Cauchy problems, we obtain well-posedness of the problem and describe the asymptotic profile of the solutions under certain natural conditions on the network. We further apply our theory to a model used for air traffic flow management.
Plasmonic waves allow perfect transmission through sub-wavelength metallic gratings
Guy Bouchitté and  Ben Schweizer
2013, 8(4): 857-878 doi: 10.3934/nhm.2013.8.857 +[Abstract](74) +[PDF](505.9KB)
We investigate the transmission properties of a metallic layer with narrow slits. Recent measurements and numerical calculations concerning the light transmission through metallic sub-wavelength structures suggest that an unexpectedly high transmission coefficient is possible. We analyze the time harmonic Maxwell's equations in the $H$-parallel case for a fixed incident wavelength. Denoting by $\eta>0$ the typical size of the complex structure, effective equations describing the limit $\eta\to 0$ are derived. For metallic permittivities with negative real part, plasmonic waves can be excited on the surfaces of the channels. When these waves are in resonance with the height of the layer, the result can be perfect transmission through the layer.
On the passage from atomistic systems to nonlinear elasticity theory for general multi-body potentials with p-growth
Julian Braun and  Bernd Schmidt
2013, 8(4): 879-912 doi: 10.3934/nhm.2013.8.879 +[Abstract](32) +[PDF](527.4KB)
We derive continuum limits of atomistic models in the realm of nonlinear elasticity theory rigorously as the interatomic distances tend to zero. In particular we obtain an integral functional acting on the deformation gradient in the continuum theory which depends on the underlying atomistic interaction potentials and the lattice geometry. The interaction potentials to which our theory applies are general finite range models on multilattices which in particular can also account for multi-pole interactions and bond-angle dependent contributions. Furthermore, we discuss the applicability of the Cauchy-Born rule. Our class of limiting energy densities consists of general quasiconvex functions and the class of linearized limiting energies consistent with the Cauchy-Born rule consists of general quadratic forms not restricted by the Cauchy relations.
Homogenization of high-contrast and non symmetric conductivities for non periodic columnar structures
Mohamed Camar-Eddine and  Laurent Pater
2013, 8(4): 913-941 doi: 10.3934/nhm.2013.8.913 +[Abstract](24) +[PDF](637.2KB)
In this paper we determine, in dimension three, the effective conductivities of non periodic and high-contrast two-phase cylindrical composites, placed in a constant magnetic field, without any assumption on the geometry of their cross sections. Our method, in the spirit of the H-convergence of Murat-Tartar, is based on a compactness result and the cylindrical nature of the microstructure. The homogenized laws we obtain extend those of the periodic fibre-reinforcing case of [17] to the case of periodic and non periodic composites with more general transversal geometries.
Global existence and asymptotic behavior of measure valued solutions to the kinetic Kuramoto--Daido model with inertia
Young-Pil Choi , Seung-Yeal Ha and  Seok-Bae Yun
2013, 8(4): 943-968 doi: 10.3934/nhm.2013.8.943 +[Abstract](61) +[PDF](473.7KB)
We present the global existence and long-time behavior of measure-valued solutions to the kinetic Kuramoto--Daido model with inertia. For the global existence of measure-valued solutions, we employ a Neunzert's mean-field approach for the Vlasov equation to construct approximate solutions. The approximate solutions are empirical measures generated by the solution to the Kuramoto--Daido model with inertia, and we also provide an a priori local-in-time stability estimate for measure-valued solutions in terms of a bounded Lipschitz distance. For the asymptotic frequency synchronization, we adopt two frameworks depending on the relative strength of inertia and show that the diameter of the projected frequency support of the measure-valued solutions exponentially converge to zero.
Convergence of vanishing capillarity approximations for scalar conservation laws with discontinuous fluxes
Giuseppe Maria Coclite , Lorenzo di Ruvo , Jan Ernest and  Siddhartha Mishra
2013, 8(4): 969-984 doi: 10.3934/nhm.2013.8.969 +[Abstract](33) +[PDF](425.8KB)
Flow of two phases in a heterogeneous porous medium is modeled by a scalar conservation law with a discontinuous coefficient. As solutions of conservation laws with discontinuous coefficients depend explicitly on the underlying small scale effects, we consider a model where the relevant small scale effect is dynamic capillary pressure. We prove that the limit of vanishing dynamic capillary pressure exists and is a weak solution of the corresponding scalar conservation law with discontinuous coefficient. A robust numerical scheme for approximating the resulting limit solutions is introduced. Numerical experiments show that the scheme is able to approximate interesting solution features such as propagating non-classical shock waves as well as discontinuous standing waves efficiently.
Structured first order conservation models for pedestrian dynamics
Dirk Hartmann and  Isabella von Sivers
2013, 8(4): 985-1007 doi: 10.3934/nhm.2013.8.985 +[Abstract](47) +[PDF](1467.6KB)
In this contribution, we revisit multiple first order macroscopic modelling approaches to pedestrian flows and computationally compare the results with a microscopic approach to pedestrian dynamics. We find that widely used conservation schemes show significantly different results than microscopic models. Thus, we propose to adopt on a macroscopic level a structured continuum model. The approach basically relies on fundamental diagrams - the relationship between fluxes and local densities - as well as the explicit consideration of individual velocities, thus showing similarities to generalised kinetic models. The macroscopic model is outlined in detail and shows a significantly better agreement with microscopic pedestrian models. The increased realism, important for safety relevant real life applications, is underlined considering several scenarios.
Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part I
Chaoqun Huang and  Nung Kwan Yip
2013, 8(4): 1009-1034 doi: 10.3934/nhm.2013.8.1009 +[Abstract](27) +[PDF](655.5KB)
We consider a singularly perturbed bistable reaction diffusion equation in a one-dimensional spatially degenerate inhomogeneous media. Degeneracy arises due to the choice of spatial inhomogeneity from some well-known class of normal forms or universal unfoldings. By means of a bilinear double well potential, we explicitly demonstrate the similarities and discrepancies between the bifurcation phenomena of the reaction diffusion equation and the limiting problem. The former is described by the location of the transition layer while the latter by the zeros of the spatial inhomogeneity function. Our result is the first which considers simultaneously the effects of singular perturbation, spatial inhomogeneity and bifurcation phenomena. (Part II [9] of this series analyzes the pitch-fork bifurcation for a general smooth double well potential where precise asymptotics and spectral analysis are needed.)

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