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Networks & Heterogeneous Media

2008 , Volume 3 , Issue 4

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Modelling and simulation of fires in tunnel networks
Ingenuin Gasser and Marcus Kraft
2008, 3(4): 691-707 doi: 10.3934/nhm.2008.3.691 +[Abstract](459) +[PDF](402.5KB)
Abstract:
A known model for describing tunnel fires is extended to the case of tunnel networks. Physically motivated coupling conditions are formulated. A numerical realisation of these conditions and of the full network problem is presented. Finally, numerical simulation of realistic tunnel fires in networks are performed.
Vertex control of flows in networks
Klaus-Jochen Engel, Marjeta Kramar Fijavž, Rainer Nagel and Eszter Sikolya
2008, 3(4): 709-722 doi: 10.3934/nhm.2008.3.709 +[Abstract](409) +[PDF](196.8KB)
Abstract:
We study a transport equation in a network and control it in a single vertex. We describe all possible reachable states and prove a criterion of Kalman type for those vertices in which the problem is maximally controllable. The results are then applied to concrete networks to show the complexity of the problem.
Stability and Riesz basis property of a star-shaped network of Euler-Bernoulli beams with joint damping
Gen Qi Xu and Siu Pang Yung
2008, 3(4): 723-747 doi: 10.3934/nhm.2008.3.723 +[Abstract](485) +[PDF](288.1KB)
Abstract:
In this paper we study a star-shaped network of Euler-Bernoulli beams, in which a new geometric condition at the common node is imposed. For the network, we give a method to assert whether or not the system is asymptotically stable. In addition, by spectral analysis of the system operator, we prove that there exists a sequence of its root vectors that forms a Riesz basis with parentheses for the Hilbert state space.
Large time behavior of nonlocal aggregation models with nonlinear diffusion
Martin Burger and Marco Di Francesco
2008, 3(4): 749-785 doi: 10.3934/nhm.2008.3.749 +[Abstract](533) +[PDF](412.0KB)
Abstract:
The aim of this paper is to establish rigorous results on the large time behavior of nonlocal models for aggregation, including the possible presence of nonlinear diffusion terms modeling local repulsions. We show that, as expected from the practical motivation as well as from numerical simulations, one obtains concentrated densities (Dirac $\delta$ distributions) as stationary solutions and large time limits in the absence of diffusion. In addition, we provide a comparison for aggregation kernels with infinite respectively finite support. In the first case, there is a unique stationary solution corresponding to concentration at the center of mass, and all solutions of the evolution problem converge to the stationary solution for large time. The speed of convergence in this case is just determined by the behavior of the aggregation kernels at zero, yielding either algebraic or exponential decay or even finite time extinction. For kernels with finite support, we show that an infinite number of stationary solutions exist, and solutions of the evolution problem converge only in a measure-valued sense to the set of stationary solutions, which we characterize in detail.
Moreover, we also consider the behavior in the presence of nonlinear diffusion terms, the most interesting case being the one of small diffusion coefficients. Via the implicit function theorem we give a quite general proof of a rather natural assertion for such models, namely that there exist stationary solutions that have the form of a local peak around the center of mass. Our approach even yields the order of the size of the support in terms of the diffusion coefficients.
All these results are obtained via a reformulation of the equations considered using the Wasserstein metric for probability measures, and are carried out in the case of a single spatial dimension.
Asymptotic analysis of a simple model of fluid-structure interaction
Serge Nicaise and Cristina Pignotti
2008, 3(4): 787-813 doi: 10.3934/nhm.2008.3.787 +[Abstract](436) +[PDF](310.3KB)
Abstract:
This paper is devoted to the asymptotic analysis of simple models of fluid-structure interaction, namely a system between the heat and wave equations coupled via some transmission conditions at the interface. The heat part induces the dissipation of the full system. Here we are interested in the behavior of the model when the thickness of the heat part and/or the heat diffusion coefficient go to zero or to infinity. The limit problem is a wave equation with a boundary condition at the interface, this boundary condition being different according to the limit of the above mentioned parameters. It turns out that some limit problems are dissipative but some of them are non dissipative or their behavior is unknown.
An existence result for the sandpile problem on flat tables with walls
Graziano Crasta and Stefano Finzi Vita
2008, 3(4): 815-830 doi: 10.3934/nhm.2008.3.815 +[Abstract](392) +[PDF](443.0KB)
Abstract:
We derive an existence result for solutions of a differential system which characterizes the equilibria of a particular model in granular matter theory, the so-called partially open table problem for growing sandpiles. Such result generalizes a recent theorem of [6] established for the totally open table problem. Here, due to the presence of walls at the boundary, the surface flow density at the equilibrium may result no more continuous nor bounded, and its explicit mathematical characterization is obtained by domain decomposition techniques. At the same time we show how these solutions can be numerically computed as stationary solutions of a dynamical two-layer model for growing sandpiles and we present the results of some simulations.
The homogenized model of small oscillations of complex fluids
M. Berezhnyi, L. Berlyand and Evgen Khruslov
2008, 3(4): 831-862 doi: 10.3934/nhm.2008.3.831 +[Abstract](402) +[PDF](411.4KB)
Abstract:
We consider the system of equations that describes small non-stationary motions of viscous incompressible fluid with a large number of small rigid interacting particles. This system is a microscopic mathematical model of complex fluids such as colloidal suspensions, polymer solutions etc. We suppose that the system of particles depends on a small parameter $\varepsilon$ in such a way that the sizes of particles are of order $\varepsilon^{3}$, the distances between the nearest particles are of order $\varepsilon$, and the stiffness of the interaction force is of order $\varepsilon^{2}$.
We study the asymptotic behavior of the microscopic model as $\varepsilon\rightarrow 0$ and obtain the homogenized equations that can be considered as a macroscopic model of diluted solutions of interacting colloidal particles.
Analysis of a reaction-diffusion system modeling predator-prey with prey-taxis
Mostafa Bendahmane
2008, 3(4): 863-879 doi: 10.3934/nhm.2008.3.863 +[Abstract](385) +[PDF](218.9KB)
Abstract:
In this paper, we consider a system of nonlinear partial differential equations modeling the Lotka Volterra interactions of preys and actively moving predators with prey-taxis and spatial diffusion. The interaction between predators are modelized by the statement of a food pyramid condition. We establish the existence of weak solutions by using Schauder fixed-point theorem and uniqueness via duality technique. This paper is a generalization of the results obtained in [2].

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