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Networks & Heterogeneous Media

2009 , Volume 4 , Issue 1

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Self--motion of camphor discs. model and analysis
Xinfu Chen , Shin-Ichiro Ei and  Masayasu Mimura
2009, 4(1): 1-18 doi: 10.3934/nhm.2009.4.1 +[Abstract](59) +[PDF](253.1KB)
In the present paper, a model describing the self-motion of a camphor disc on water is proposed. The stability of a standing camphor disc is investigated by analyzing the model equation, and a pitchfork type bifurcation diagram of a traveling spot is shown. Multiple camphor discs are also treated by the model equations, and the repulsive interaction of spots is discussed.
Feedback stabilization of a coupled string-beam system
Kaïs Ammari , Mohamed Jellouli and  Michel Mehrenberger
2009, 4(1): 19-34 doi: 10.3934/nhm.2009.4.19 +[Abstract](51) +[PDF](224.9KB)
We consider a stabilization problem for a coupled string-beam system. We prove some decay results of the energy of the system. The method used is based on the methodology introduced in Ammari and Tucsnak [2] where the exponential and weak stability for the closed loop problem is reduced to a boundedness property of the transfer function of the associated open loop system. Morever, we prove, for the same model but with two control functions, independently of the length of the beam that the energy decay with a polynomial rate for all regular initial data. The method used, in this case, is based on a frequency domain method and combine a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent.
A uniformly second order numerical method for the one-dimensional discrete-ordinate transport equation and its diffusion limit with interface
Shi Jin , Min Tang and  Houde Han
2009, 4(1): 35-65 doi: 10.3934/nhm.2009.4.35 +[Abstract](46) +[PDF](340.0KB)
In this paper, we propose a uniformly second order numerical method for the discete-ordinate transport equation in the slab geometry in the diffusive regimes with interfaces. At the interfaces, the scattering coefficients have discontinuities, so suitable interface conditions are needed to define the unique solution. We first approximate the scattering coefficients by piecewise constants determined by their cell averages, and then obtain the analytic solution at each cell, using which to piece together the numerical solution with the neighboring cells by the interface conditions. We show that this method is asymptotic-preserving, which preserves the discrete diffusion limit with the correct interface condition. Moreover, we show that our method is quadratically convergent uniformly in the diffusive regime, even with the boundary layers. This is 1) the first sharp uniform convergence result for linear transport equations in the diffusive regime, a problem that involves both transport and diffusive scales; and 2) the first uniform convergence valid up to the boundary even if the boundary layers exist, so the boundary layer does not need to be resolved numerically. Numerical examples are presented to justify the uniform convergence.
G1/S transition and cell population dynamics
Fadia Bekkal-Brikci , Giovanna Chiorino and  Khalid Boushaba
2009, 4(1): 67-90 doi: 10.3934/nhm.2009.4.67 +[Abstract](43) +[PDF](365.5KB)
In this paper we present a model connecting the state of molecular components during the cell cycle at the individual level to the population dynamic. The complexes Cyclin E/CDK2 are good markers of the cell state in its cycle. In this paper we focus on the first transition phase of the cell cycle ($S-G_{2}-M$) where the complexe Cyclin E/CDK2 has a key role in this transition. We give a simple system of differential equations to represent the dynamic of the Cyclin E/CDK2 amount during the cell cycle, and couple it with a cell population dynamic in such way our cell population model is structured by cell age and the amount of Cyclin E/CDK2 with two compartments: cells in the G1 phase and cells in the remainder of the cell cycle ($S-G_{2}-M$). A cell transits from the G1 phase to the S phase when Cyclin E/CDK2 reaches a threshold, which allow us to take into account the variability in the timing of G1/S transition. Then the cell passes through $S-G_{2}-M$ phases and divides with the assumption of unequal division among daughter cells of the final Cyclin E/CDK2 amount. The existence and the asymptotic behavior of the solution of the model is analyzed.
Tailored finite point method for the interface problem
Zhongyi Huang
2009, 4(1): 91-106 doi: 10.3934/nhm.2009.4.91 +[Abstract](36) +[PDF](340.9KB)
In this paper, we propose a tailored-finite-point method for a numerical simulation of the second order elliptic equation with discontinuous coefficients. Our finite point method has been tailored to some particular properties of the problem, then we can get the approximate solution with the same behaviors as that of the exact solution very naturally. Especially, in one-dimensional case, when the coefficients are piecewise linear functions, we can get the exact solution with only one point in each subdomain. Furthermore, the stability analysis and the uniform convergence analysis in the energy norm are proved. On the other hand, our computational complexity is only $\O(N)$ for $N$ discrete points. We also extend our method to two-dimensional problems.
On fluido-dynamic models for urban traffic
Mauro Garavello and  Benedetto Piccoli
2009, 4(1): 107-126 doi: 10.3934/nhm.2009.4.107 +[Abstract](42) +[PDF](263.5KB)
The aim of this paper is to address the following questions: which models, among fluido-dynamic ones, are more appropriate to describe urban traffic? While a rich debate was developed for the complicate dynamics of highway traffic, some basic problems of urban traffic are not always appropriately discussed. We analyze many recent, and less recent, models focusing on three basic properties. The latter are necessary to reproduce correctly queue formation at lights and junctions, and their backward propagation on an urban network.
Homogenization of interfacial energies and construction of plane-like minimizers in periodic media through a cell problem
Antonin Chambolle and  Gilles Thouroude
2009, 4(1): 127-152 doi: 10.3934/nhm.2009.4.127 +[Abstract](41) +[PDF](347.8KB)
We consider the homogenization of a periodic interfacial energy, such as considered in recents papers by Caffarelli and De La Llave [14], or Dirr, Lucia and Novaga [16]. In particular, we include the case where an external forcing field (which is unbounded in the limit) is present, and suggest two different ways to take care of this additional perturbation. We provide a proof of a $\Gamma$-limit, however, we also observe that thanks to the coarea formula, in many cases such a result is already known in the framework of $BV$ homogenization. This leads to an interesting new construction for the plane-like minimizers in periodic media of Caffarelli and De La Llave, through a cell problem.
Korn inequalities on thin periodic structures
V. V. Zhikov and  S. E. Pastukhova
2009, 4(1): 153-175 doi: 10.3934/nhm.2009.4.153 +[Abstract](38) +[PDF](454.1KB)
We prove Korn-type inequalities for thin periodic structures of period $\varepsilon$ and thickness $\varepsilon h(\varepsilon)$, where $h(\varepsilon)\to 0$ as $\varepsilon\to 0$, among which there are plane grids, spatial rod and box structures. These inequalities are important in homogenization of corresponding elasticity problems.

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