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Volume 1, 2006

Networks & Heterogeneous Media

2006 , Volume 1 , Issue 1

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Benedetto Piccoli
2006, 1(1): i-ii doi: 10.3934/nhm.2006.1.1i +[Abstract](26) +[PDF](29.6KB)
To start a new journal in a fast growing scientific panorama is a serious challenge. We decided to start this new adventure because we felt the necessity of having an applied math journal covering an area of great interest and experiencing a big expansion.

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HTTP turbulence
François Baccelli , Augustin Chaintreau , Danny De Vleeschauwer and  David R. McDonald
2006, 1(1): 1-40 doi: 10.3934/nhm.2006.1.1 +[Abstract](50) +[PDF](1344.4KB)
In this paper, we consider a set of HTTP flows using TCP over a common drop-tail link to download files. After each download, a flow waits for a random think time before requesting the download of another file, whose size is also random. When a flow is active its throughput is increasing with time according to the additive increase rule, but if it su®ers losses created when the total transmission rate of the flows exceeds the link rate, its transmission rate is decreased. The throughput obtained by a °ow, and the consecutive time to download one file are then given as the consequence of the interaction of all the flows through their total transmission rate and the link's behavior.
    We study the mean-field model obtained by letting the number of flows go to infinity. This mean-field limit may have two stable regimes: one with- out congestion in the link, in which the density of transmission rate can be explicitly described, the other one with periodic congestion epochs, where the inter-congestion time can be characterized as the solution of a fixed point equation, that we compute numerically, leading to a density of transmission rate given by as the solution of a Fredholm equation. It is shown that for certain values of the parameters (more precisely when the link capacity per user is not significantly larger than the load per user), each of these two stable regimes can be reached depending on the initial condition. This phenomenon can be seen as an analogue of turbulence in fluid dynamics: for some initial conditions, the transfers progress in a fluid and interaction-less way; for others, the connections interact and slow down because of the resulting fluctuations, which in turn perpetuates interaction forever, in spite of the fact that the load per user is less than the capacity per user. We prove that this phenomenon is present in the Tahoe case and both the numerical method that we develop and simulations suggest that it is also be present in the Reno case. It translates into a bi-stability phenomenon for the finite population model within this range of parameters.
Gas flow in pipeline networks
Mapundi K. Banda , Michael Herty and  Axel Klar
2006, 1(1): 41-56 doi: 10.3934/nhm.2006.1.41 +[Abstract](73) +[PDF](3881.2KB)
We introduce a model for gas flow in pipeline networks based on the isothermal Euler equations. We model the intersection of multiple pipes by posing an additional assumption on the pressure at the interface. We give a method to obtain solutions to the gas network problem and present numerical results for sample networks.
Numerical approximations of a traffic flow model on networks
Gabriella Bretti , Roberto Natalini and  Benedetto Piccoli
2006, 1(1): 57-84 doi: 10.3934/nhm.2006.1.57 +[Abstract](103) +[PDF](552.6KB)
We consider a mathematical model for fluid-dynamic flows on networks which is based on conservation laws. Road networks are considered as graphs composed by arcs that meet at some junctions. The crucial point is represented by junctions, where interactions occur and the problem is underdetermined. The approximation of scalar conservation laws along arcs is carried out by using conservative methods, such as the classical Godunov scheme and the more recent discrete velocities kinetic schemes with the use of suitable boundary conditions at junctions. Riemann problems are solved by means of a simulation algorithm which proceeds processing each junction. We present the algorithm and its application to some simple test cases and to portions of urban network.
Phase and anti-phase boundaries in binary discrete systems: a variational viewpoint
Roberto Alicandro , Andrea Braides and  Marco Cicalese
2006, 1(1): 85-107 doi: 10.3934/nhm.2006.1.85 +[Abstract](68) +[PDF](285.9KB)
We provide a variational description of nearest-neighbours and next-to-nearest neighbours binary lattice systems. By studying the $\Gamma$-limit of proper scaling of the energies of the systems, we highlight phase and anti-phase boundary phenomena and show how they depend on the geometry of the lattice.
A direct approach to numerical homogenization in finite elasticity
Antoine Gloria Cermics
2006, 1(1): 109-141 doi: 10.3934/nhm.2006.1.109 +[Abstract](41) +[PDF](494.2KB)
We describe, analyze, and test a direct numerical approach to a homogenized problem in nonlinear elasticity at finite strain. The main advantage of this approach is that it does not modify the overall structure of standard softwares in use for computational elasticity. Our analysis includes a convergence result for a general class of energy densities and an error estimate in the convex case. We relate this approach to the multiscale finite element method and show our analysis also applies to this method. Microscopic buck- ling and macroscopic instabilities are numerically investigated. The application of our approach to some numerical tests on an idealized rubber foam is also presented. For consistency a short review of the homogenization theory in nonlinear elasticity is provided.
Numerical study of a domain decomposition method for a two-scale linear transport equation
Xu Yang , François Golse , Zhongyi Huang and  Shi Jin
2006, 1(1): 143-166 doi: 10.3934/nhm.2006.1.143 +[Abstract](55) +[PDF](495.5KB)
We perform a numerical study on a domain decomposition method proposed in [13] for the linear transport equation between a diffusive and a non-diffusive region. This method avoids iterating the diffusion and transport solutions as in a typical domain decomposition method. Our numerical results, in both one and two space dimensions, confirm the theoretical analysis of [13]. We also provide an improved second order method that provides a more accurate numerical solution than that proposed in [13].
The Green's functions for the Broadwell Model in a half space problem
Chiu-Ya Lan , Huey-Er Lin and  Shih-Hsien Yu
2006, 1(1): 167-183 doi: 10.3934/nhm.2006.1.167 +[Abstract](62) +[PDF](292.5KB)
We study an initial boundary value problem for the Broadwell model with a supersonic physical boundary. The Green's function for an initial value problem is constructed and its detailed pointwise structure is obtained through the novel decompositions introduced in [8]. With the Green's function for initial value problem and energy estimates together, a new approach to convert a priori $L^2$-boundary data into $L^\infty$ boundary data is established for the Broadwell model. The Green's function for an initial boundary value problem is obtained. Finally, a nonlinearly time-asymptotic stability of an equilibrium state is proved.
Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue
Mostafa Bendahmane and  Kenneth H. Karlsen
2006, 1(1): 185-218 doi: 10.3934/nhm.2006.1.185 +[Abstract](36) +[PDF](414.3KB)
We prove well-posedness (existence and uniqueness) results for a class of degenerate reaction-diffusion systems. A prototype system belonging to this class is provided by the bidomain model, which is frequently used to study and simulate electrophysiological waves in cardiac tissue. The existence result, which constitutes the main thrust of this paper, is proved by means of a nondegenerate approximation system, the Faedo-Galerkin method, and the compactness method.
Equilibria and stability analysis of a branched metabolic network with feedback inhibition
Yacine Chitour , Frédéric Grognard and  Georges Bastin
2006, 1(1): 219-239 doi: 10.3934/nhm.2006.1.219 +[Abstract](51) +[PDF](382.0KB)
This paper deals with the analysis of a metabolic network with feedback inhibition. The considered system is an acyclic network of mono-molecular enzymatic reactions in which metabolites can act as feedback regulators on enzymes located "at the beginning" of their own pathway, and in which one metabolite is the root of the whole network. We show, under mild assumptions, the uniqueness of the equilibrium. We then show that this equilibrium is globally attractive if we impose conditions on the kinetic parameters of the metabolic reactions. Finally, when these conditions are not satisfied, we show, with a specific fourth-order example, that the equilibrium may become unstable with an attracting limit cycle.

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