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Networks & Heterogeneous Media

2007 , Volume 2 , Issue 2

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Self-organized network flows
Dirk Helbing , Jan Siegmeier and  Stefan Lämmer
2007, 2(2): 193-210 doi: 10.3934/nhm.2007.2.193 +[Abstract](52) +[PDF](545.0KB)
A model for traffic flow in street networks or material flows in supply networks is presented, that takes into account the conservation of cars or materials and other significant features of traffic flows such as jam formation, spillovers, and load-dependent transportation times. Furthermore, conflicts or coordination problems of intersecting or merging flows are considered as well. Making assumptions regarding the permeability of the intersection as a function of the conflicting flows and the queue lengths, we find self-organized oscillations in the flows similar to the operation of traffic lights.
A new model for contact angle hysteresis
Antonio DeSimone , Natalie Grunewald and  Felix Otto
2007, 2(2): 211-225 doi: 10.3934/nhm.2007.2.211 +[Abstract](62) +[PDF](329.8KB)
We present a model which explains several experimental observations relating contact angle hysteresis with surface roughness. The model is based on the balance between released capillary energy and dissipation associated with motion of the contact line: it describes the stick–slip behavior of drops on a rough surface using ideas similar to those employed in dry friction, elasto–plasticity and fracture mechanics. The main results of our analysis are formulas giving the interval of stable contact angles as a function of the surface roughness. These formulas show that the difference between advancing and receding angles is much larger for a drop in complete contact with the substrate (Wenzel drop) than for one whose cavities are filled with air (Cassie-Baxter drop). This fact is used as the key tool to interpret the experimental evidence.
A second order model of road junctions in fluid models of traffic networks
Bertrand Haut and  Georges Bastin
2007, 2(2): 227-253 doi: 10.3934/nhm.2007.2.227 +[Abstract](48) +[PDF](386.7KB)
This article deals with the modeling of junctions in a road network from a macroscopic point of view. After reviewing the Aw & Rascle second order model, a compatible junction model is proposed. The properties of this model and particularly the stability are analyzed. It turns out that this model presents physically acceptable solutions, is able to represent the capacity drop phenomenon and can be used to simulate the traffic evolution on a network.
Asymptotic analysis of a perturbed parabolic problem in a thick junction of type 3:2:2
Ciro D’Apice , Umberto De Maio and  T. A. Mel'nyk
2007, 2(2): 255-277 doi: 10.3934/nhm.2007.2.255 +[Abstract](37) +[PDF](301.8KB)
We consider a perturbed initial/boundary-value problem for the heat equation in a thick multi-structure $\Omega_{\varepsilon}$ which is the union of a domain $\Omega_0$ and a large number $N$ of $\varepsilon-$periodically situated thin rings with variable thickness of order $\varepsilon = \mathcal{O}(N^{-1}).$ The following boundary condition $\partial_{\nu}u_{\varepsilon} + \varepsilon^{\alpha} k_0 u_{\varepsilon}= \varepsilon^{\beta} g_{\varepsilon}$ is given on the lateral boundaries of the thin rings; here the parameters $\alpha$ and $\beta$ are greater than or equal $1.$ The asymptotic analysis of this problem for different values of the parameters $\alpha$ and $\beta$ is made as $\varepsilon\to0.$ The leading terms of the asymptotic expansion for the solution are constructed, the corresponding estimates in the Sobolev space $L^2(0,T; H^1(\Omega_{\varepsilon}))$ are obtained and the convergence theorem is proved with minimal conditions for the right-hand sides.
Ideally soft nematic elastomers
M. Silhavý
2007, 2(2): 279-311 doi: 10.3934/nhm.2007.2.279 +[Abstract](45) +[PDF](398.7KB)
The paper examines a class of energies $W$ of nematic elastomers that exhibit ideally soft behavior. These are generalizations of the neo-classical energy function proposed by Bladon, Terentjev & Warner [7]. The effective energy (quasiconvexification) of $W$ is calculated for a large subclass of considered energies. Within the subclass, the rank 1 convex, quasiconvex, and polyconvex envelopes coincide and reduce to the largest function below $W$ that satisfies the Baker–Ericksen inequalities. Compressible cases are included. The effective energy displays three regimes: one fluid-like, one partially fluid-like and one hard, as established by DeSimone & Dolzmann [20] for the energy function of Bladon, Terentjev & Warner. Ideally soft deformation modes are shown to arise.
Optimal control problems on stratified domains
Alberto Bressan and  Yunho Hong
2007, 2(2): 313-331 doi: 10.3934/nhm.2007.2.313 +[Abstract](47) +[PDF](260.0KB)
We consider a class of optimal control problems defined on a stratified domain. Namely, we assume that the state space $\mathbb{R}^N$ admits a stratification as a disjoint union of finitely many embedded submanifolds $\mathcal{M}_i$. The dynamics of the system and the cost function are Lipschitz continuous restricted to each submanifold. We provide conditions which guarantee the existence of an optimal solution, and study sufficient conditions for optimality. These are obtained by proving a uniqueness result for solutions to a corresponding Hamilton-Jacobi equation with discontinuous coefficients, describing the value function. Our results are motivated by various applications, such as minimum time problems with discontinuous dynamics, and optimization problems constrained to a bounded domain, in the presence of an additional overflow cost at the boundary.
Modeling cell movement in anisotropic and heterogeneous network tissues
A. Chauviere , T. Hillen and  L. Preziosi
2007, 2(2): 333-357 doi: 10.3934/nhm.2007.2.333 +[Abstract](49) +[PDF](2838.8KB)
Cell motion and interaction with the extracellular matrix is studied deriving a kinetic model and considering its diffusive limit. The model takes into account the chemotactic and haptotactic effects, and obtains friction as a result of the interactions between cells and between cells and the fibrous environment. The evolution depends on the fibre distribution, as cells preferentially move along the fibre direction and tend to cleave and remodel the extracellular matrix when their direction of motion is not aligned with the fibre direction. Simulations are performed to describe the behavior of an ensemble of cells under the action of a chemotactic field and in the presence of heterogeneous and anisotropic fibre networks.
Nonlinear transmission problems for quasilinear diffusion systems
F. R. Guarguaglini and  R. Natalini
2007, 2(2): 359-381 doi: 10.3934/nhm.2007.2.359 +[Abstract](35) +[PDF](262.2KB)
We study degenerate quasilinear parabolic systems in two different domains, which are connected by a nonlinear transmission condition at their interface. For a large class of models, including those modeling pollution aggression on stones and chemotactic movements of bacteria, we prove global existence, uniqueness and stability of the solutions.

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