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Networks & Heterogeneous Media

2010 , Volume 5 , Issue 1

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Improving on computation of homogenized coefficients in the periodic and quasi-periodic settings
Xavier Blanc and Claude Le Bris
2010, 5(1): 1-29 doi: 10.3934/nhm.2010.5.1 +[Abstract](88) +[PDF](333.4KB)
In quasi-periodic homogenization of elliptic equations or nonlinear periodic homogenization of systems, the cell problem must be in general set on the whole space. Numerically computing the homogenization coefficient therefore implies a truncation error, due to the fact that the problem is approximated on a bounded, large domain. We present here an approach that improves the rate of convergence of this approximation.
A particle system in interaction with a rapidly varying environment: Mean field limits and applications
Charles Bordenave, David R. McDonald and Alexandre Proutière
2010, 5(1): 31-62 doi: 10.3934/nhm.2010.5.31 +[Abstract](107) +[PDF](374.3KB)
We study an interacting particle system whose dynamics depends on an interacting random environment. As the number of particles grows large, the transition rate of the particles slows down (perhaps because they share a common resource of fixed capacity). The transition rate of a particle is determined by its state, by the empirical distribution of all the particles and by a rapidly varying environment. The transitions of the environment are determined by the empirical distribution of the particles. We prove the propagation of chaos on the path space of the particles and establish that the limiting trajectory of the empirical measure of the states of the particles satisfies a deterministic differential equation. This deterministic differential equation involves the time averages of the environment process.
   We apply the results on particle systems to understand the behavior of computer networks where users access a shared resource using a distributed random Medium Access Control (MAC) algorithm. MAC algorithms are used in all Local Area Network (LAN), and have been notoriously difficult to analyze. Our analysis allows us to provide simple and explicit expressions of the network performance under such algorithms.
Flows in porous media with erosion of the solid matrix
Leda Bucciantini, Angiolo Farina and Antonio Fasano
2010, 5(1): 63-95 doi: 10.3934/nhm.2010.5.63 +[Abstract](105) +[PDF](491.1KB)
We consider the flow of an incompressible Newtonian fluid through an idealized porous medium consisting of an array of identical solid symmetric lamellae, whose profile varies in space and time due to a stress induced erosion process. The focus is on the influence of mass exchange between solid and fluid on the macroscopic flow. By means of the upscaling procedure illustrated in [6] we derive the governing system of equations for the macroscopic flow, encompassing various physical situations. We show that Darcy's law no longer applies in the classical sense. The corresponding mathematical problem turns out to be surprisingly complicated. Existence and uniqueness are proved. Numerical simulations are presented.
Quasistatic evolution for Cam-Clay plasticity: The spatially homogeneous case
Gianni Dal Maso and Francesco Solombrino
2010, 5(1): 97-132 doi: 10.3934/nhm.2010.5.97 +[Abstract](98) +[PDF](457.2KB)
We study the spatially uniform case of the quasistatic evolution in Cam-Clay plasticity, a relevant example of small strain nonassociative elastoplasticity. Introducing a viscous approximation, the problem reduces to determine the limit behavior of the solutions of a singularly perturbed system of ODE's in a finite dimensional Banach space. Depending on the sign of two explicit scalar indicators, we see that the limit dynamics presents, under quite generic assumptions, the alternation of three possible regimes: the elastic regime, when the limit equation is just the equation of linearized elasticity; the slow dynamics, when the stress evolves smoothly on the yield surface and plastic flow is produced; the fast dynamics, which may happen only in the softening regime, when viscous solutions exhibit a jump determined by the heteroclinic orbit of an auxiliary system. We give an iterative procedure to construct a viscous solution.
Large time behavior for the IBVP of the 3-D Nishida's model
Shijin Deng
2010, 5(1): 133-142 doi: 10.3934/nhm.2010.5.133 +[Abstract](63) +[PDF](157.1KB)
In this paper we investigate an initial boundary value problem (IBVP) for the Nishda's model in 3-dimensional space with a forward moving physical boundary. It is shown that the solution converges to zero with an exponential rate by energy estimates.
The coolest path problem
Martin Frank, Armin Fügenschuh, Michael Herty and Lars Schewe
2010, 5(1): 143-162 doi: 10.3934/nhm.2010.5.143 +[Abstract](95) +[PDF](548.6KB)
We introduce the coolest path problem, which is a mixture of two well-known problems from distinct mathematical fields. One of them is the shortest path problem from combinatorial optimization. The other is the heat conduction problem from the field of partial differential equations. Together, they make up a control problem, where some geometrical object traverses a digraph in an optimal way, with constraints on intermediate or the final state. We discuss some properties of the problem and present numerical solution techniques. We demonstrate that the problem can be formulated as a linear mixed-integer program. Numerical solutions can thus be achieved within one hour for instances with up to 70 nodes in the graph.
Existence and stability of a multidimensional scalar conservation law with discontinuous flux
Darko Mitrovic
2010, 5(1): 163-188 doi: 10.3934/nhm.2010.5.163 +[Abstract](98) +[PDF](291.4KB)
We prove the existence and stability of an entropy solution to a multidimensional scalar conservation law with discontinuous flux with no genuine nonlinearity assumptions. The proof is based on the corresponding kinetic formulation of the equation under consideration and a "smart" change of an unknown function.

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