# American Institute of Mathematical Sciences

ISSN:
1556-1801

eISSN:
1556-181X

All Issues

## Networks & Heterogeneous Media

2006 , Volume 1 , Issue 2

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2006, 1(2): 241-258 doi: 10.3934/nhm.2006.1.241 +[Abstract](235) +[PDF](401.5KB)
Abstract:
This work is concerned with some aspects of the social life of the amoebae Dictyostelium discoideum (Dd). In particular, we shall focus on the early stages of the starvation-induced aggregation of Dd cells. Under such circumstances, amoebae are known to exchange a chemical messenger (cAMP) which acts as a signal to mediate their individual behaviour. This molecule is released from aggregation centres and advances through aggregation ﬁelds, ﬁrst as circular waves and later on as spiral patterns. We shall recall below some of the basic features of this process, paying attention to the mathematical models that have been derived to account for experimental observations.
2006, 1(2): 259-274 doi: 10.3934/nhm.2006.1.259 +[Abstract](229) +[PDF](1124.1KB)
Abstract:
Under consideration is the finnite-size scaling of effective thermoelastic properties of random microstructures from a Statistical Volume Element (SVE) to a Representative Volume Element (RVE), without invoking any periodic structure assumptions, but only assuming the microstructure's statistics to be spatially homogeneous and ergodic. The SVE is set up on a mesoscale, i.e. any scale finite relative to the microstructural length scale. The Hill condition generalized to thermoelasticity dictates uniform Neumann and Dirichlet boundary conditions, which, with the help of two variational principles, lead to scale dependent hierarchies of mesoscale bounds on effective (RVE level) properties: thermal expansion and stress coefficients, effective stiffness, and specific heats. Due to the presence of a non-quadratic term in the energy formulas, the mesoscale bounds for the thermal expansion are more complicated than those for the stiffness tensor and the heat capacity. To quantitatively assess the scaling trend towards the RVE, the hierarchies are computed for a planar matrix-inclusion composite, with inclusions (of circular disk shape) located at points of a planar, hard-core Poisson point field. Overall, while the RVE is attained exactly on scales infinitely large relative to the microscale, depending on the microstructural parameters, the random fluctuations in the SVE response may become very weak on scales an order of magnitude larger than the microscale, thus already approximating the RVE.
2006, 1(2): 275-294 doi: 10.3934/nhm.2006.1.275 +[Abstract](248) +[PDF](475.4KB)
Abstract:
We consider coupling conditions for the “Aw–Rascle” (AR) traffic ﬂow model at an arbitrary road intersection. In contrast with coupling conditions previously introduced in [10] and [7], all the moments of the AR system are conserved and the total ﬂux at the junction is maximized. This nonlinear optimization problem is solved completely. We show how the two simple cases of merging and diverging junctions can be extended to more complex junctions, like roundabouts. Finally, we present some numerical results.
2006, 1(2): 295-314 doi: 10.3934/nhm.2006.1.295 +[Abstract](403) +[PDF](1720.1KB)
Abstract:
We investigate coupling conditions for gas transport in networks where the governing equations are the isothermal Euler equations. We discuss intersections of pipes by considering solutions to Riemann problems. We introduce additional assumptions to obtain a solution near the intersection and we present numerical results for sample networks.
2006, 1(2): 315-336 doi: 10.3934/nhm.2006.1.315 +[Abstract](275) +[PDF](272.9KB)
Abstract:
The aim of this paper is to optimize tra±c distribution coefficients in order to maximize the trasmission speed of packets over a network. We consider a macroscopic fluidodynamic model dealing with packets flow proposed in [10], where the dynamics at nodes (routers) is decided by a routing algorithm depending on traffic distribution (and priority) coefficients. We solve the general problem for a node with m incoming and n outgoing lines and explicit the optimal parameters for the simple case of two incoming and two outgoing lines.
2006, 1(2): 337-351 doi: 10.3934/nhm.2006.1.337 +[Abstract](246) +[PDF](252.7KB)
Abstract:
We consider the initial value problem for the filtration equation in an inhomogeneous medium
$p(x)u_t = \Delta u^m, m>1$.

The equation is posed in the whole space $\mathbb R^n$ , $n \geq 2$, for $0 < t < \infty$; $p(x)$ is a positive and bounded function with a certain behaviour at infinity. We take initial data $u(x,0) = u_0(x) \geq 0$, and prove that this problem is well-posed in the class of solutions with finite "energy", that is, in the weighted space $L^1_p$, thus completing previous work of several authors on the issue. Indeed, it generates a contraction semigroup.
We also study the asymptotic behaviour of solutions in two space dimensions when $p$ decays like a non-integrable power as $|x| \rightarrow \infty$ : $p(x)$ $|x|^\alpha$ ~ $1$ with $\alpha \epsilon (0,2)$ (infinite mass medium). We show that the intermediate asymptotics is given by the unique selfsimilar solution $U_2(x, t; E)$ of the singular problem
$|x|^{- \alpha} u_t = \Delta u_m$ in $\mathbb R^2 \times \mathbb R_+$
$|x|^{- \alpha} u(x,0) = E\delta(x), E = ||u_0||_{L^1_p}$

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