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Networks and Heterogeneous Media

September 2016 , Volume 11 , Issue 3

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Varying the direction of propagation in reaction-diffusion equations in periodic media
Matthieu Alfaro and Thomas Giletti
2016, 11(3): 369-393 doi: 10.3934/nhm.2016001 +[Abstract](3545) +[PDF](487.1KB)
We consider a multidimensional reaction-diffusion equation of either ignition or monostable type, involving periodic heterogeneity, and analyze the dependence of the propagation phenomena on the direction. We prove that the (minimal) speed of the underlying pulsating fronts depends continuously on the direction of propagation, and so does its associated profile provided it is unique up to time shifts. We also prove that the spreading properties [25] are actually uniform with respect to the direction.
On the micro-to-macro limit for first-order traffic flow models on networks
Emiliano Cristiani and Smita Sahu
2016, 11(3): 395-413 doi: 10.3934/nhm.2016002 +[Abstract](4012) +[PDF](513.3KB)
Connections between microscopic follow-the-leader and macroscopic fluid-dynamics traffic flow models are already well understood in the case of vehicles moving on a single road. Analogous connections in the case of road networks are instead lacking. This is probably due to the fact that macroscopic traffic models on networks are in general ill-posed, since the conservation of the mass is not sufficient alone to characterize a unique solution at junctions. This ambiguity makes more difficult to find the right limit of the microscopic model, which, in turn, can be defined in different ways near the junctions. In this paper we show that a natural extension of the first-order follow-the-leader model on networks corresponds, as the number of vehicles tends to infinity, to the LWR-based multi-path model introduced in [4,5].
On optimization of a highly re-entrant production system
Ciro D'Apice, Peter I. Kogut and Rosanna Manzo
2016, 11(3): 415-445 doi: 10.3934/nhm.2016003 +[Abstract](3417) +[PDF](521.5KB)
We discuss the optimal control problem stated as the minimization in the $L^2$-sense of the mismatch between the actual out-flux and a demand forecast for a hyperbolic conservation law that models a highly re-entrant production system. The output of the factory is described as a function of the work in progress and the position of the switch dispatch point (SDP) where we separate the beginning of the factory employing a push policy from the end of the factory, which uses a quasi-pull policy. The main question we discuss in this paper is about the optimal choice of the input in-flux, push and quasi-pull constituents, and the position of SDP.
A weakly coupled model of differential equations for thief tracking
Simone Göttlich and Camill Harter
2016, 11(3): 447-469 doi: 10.3934/nhm.2016004 +[Abstract](3906) +[PDF](502.1KB)
In this work we introduce a novel model for the tracking of a thief moving through a road network. The modeling equations are given by a strongly coupled system of scalar conservation laws for the road traffic and ordinary differential equations for the thief evolution. A crucial point is the characterization at intersections, where the thief has to take a routing decision depending on the available local information. We develop a numerical approach to solve the thief tracking problem by combining a time-dependent shortest path algorithm with the numerical solution of the traffic flow equations. Various computational experiments are presented to describe different behavior patterns.
Osmosis for non-electrolyte solvents in permeable periodic porous media
Alexei Heintz and Andrey Piatnitski
2016, 11(3): 471-499 doi: 10.3934/nhm.2016005 +[Abstract](3517) +[PDF](506.8KB)
The paper gives a rigorous description, based on mathematical homogenization theory, for flows of solvents with not charged solute particles under osmotic pressure for periodic porous media permeable for solute particles. The effective Darcy type equations for the flow under osmotic pressure distributed within the porous media are derived. The effective Darcy law contains an additional flux term representing the osmotic pressure. Coefficients in the effective homogenized equations are related to the values of the phenomenological coefficients in the Kedem-Katchalsky formulae (2).
A note on non lower semicontinuous perimeter functionals on partitions
Annibale Magni and Matteo Novaga
2016, 11(3): 501-508 doi: 10.3934/nhm.2016006 +[Abstract](2937) +[PDF](328.5KB)
We consider isotropic non lower semicontinuous weighted perimeter functionals defined on partitions of domains in $\mathbb{R}^n$. Besides identifying a condition on the structure of the domain which ensures the existence of minimizing configurations, we describe the structure of such minima, as well as their regularity.
Evolution of spoon-shaped networks
Alessandra Pluda
2016, 11(3): 509-526 doi: 10.3934/nhm.2016007 +[Abstract](2637) +[PDF](427.4KB)
We consider a regular embedded network composed by two curves, one of them closed, in a convex and smooth domain $\Omega$. The two curves meet only at one point, forming angles of $120$ degrees. The non-closed curve has a fixed end--point on $\partial\Omega$. We study the evolution by curvature of this network. We show that the maximal time of existence is finite and depends only on the area enclosed in the initial loop, if the length of the non-closed curve stays bounded from below during the evolution. Moreover, the closed curve shrinks to a point and the network is asymptotically approaching, after dilations and extraction of a subsequence, a Brakke spoon.
The exponential decay rate of generic tree of 1-d wave equations with boundary feedback controls
Yaru Xie and Genqi Xu
2016, 11(3): 527-543 doi: 10.3934/nhm.2016008 +[Abstract](4311) +[PDF](389.5KB)
In this paper, we study the exponential decay rate of generic tree of 1-d wave equations with boundary feedback controls. For the networks, there are some results on the exponential stability, but no result on estimate of the decay rate. The present work mainly estimates the decay rate for these systems, including signal wave equation, serially connected wave equations, and generic tree of 1-d wave equations. By defining the weighted energy functional of the system, and choosing suitable weighted functions, we obtain the estimation value of decay rate of the systems.

2020 Impact Factor: 1.213
5 Year Impact Factor: 1.384
2021 CiteScore: 2.2




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