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Networks & Heterogeneous Media

2015 , Volume 10 , Issue 2

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On gradient structures for Markov chains and the passage to Wasserstein gradient flows
Karoline Disser and Matthias Liero
2015, 10(2): 233-253 doi: 10.3934/nhm.2015.10.233 +[Abstract](406) +[PDF](472.2KB)
We study the approximation of Wasserstein gradient structures by their finite-dimensional analog. We show that simple finite-volume discretizations of the linear Fokker-Planck equation exhibit the recently established entropic gradient-flow structure for reversible Markov chains. Then we reprove the convergence of the discrete scheme in the limit of vanishing mesh size using only the involved gradient-flow structures. In particular, we make no use of the linearity of the equations nor of the fact that the Fokker-Planck equation is of second order.
Conservation law models for traffic flow on a network of roads
Alberto Bressan and Khai T. Nguyen
2015, 10(2): 255-293 doi: 10.3934/nhm.2015.10.255 +[Abstract](513) +[PDF](674.2KB)
The paper develops a model of traffic flow near an intersection, where drivers seeking to enter a congested road wait in a buffer of limited capacity. Initial data comprise the vehicle density on each road, together with the percentage of drivers approaching the intersection who wish to turn into each of the outgoing roads.
    If the queue sizes within the buffer are known, then the initial-boundary value problems become decoupled and can be independently solved along each incoming road. Three variational problems are introduced, related to different kind of boundary conditions. From the value functions, one recovers the traffic density along each incoming or outgoing road by a Lax type formula.
    Conversely, if these value functions are known, then the queue sizes can be determined by balancing the boundary fluxes of all incoming and outgoing roads. In this way one obtains a contractive transformation, whose fixed point yields the unique solution of the Cauchy problem for traffic flow in an neighborhood of the intersection.
    The present model accounts for backward propagation of queues along roads leading to a crowded intersection, it achieves well-posedness for general $L^\infty $ data, and continuity w.r.t. weak convergence of the initial densities.
Stationary states in gas networks
Martin Gugat, Falk M. Hante, Markus Hirsch-Dick and Günter Leugering
2015, 10(2): 295-320 doi: 10.3934/nhm.2015.10.295 +[Abstract](441) +[PDF](557.4KB)
Pipeline networks for gas transportation often contain circles. For such networks it is more difficult to determine the stationary states than for networks without circles. We present a method that allows to compute the stationary states for subsonic pipe flow governed by the isothermal Euler equations for certain pipeline networks that contain circles. We also show that suitably chosen boundary data determine the stationary states uniquely. The construction is based upon novel explicit representations of the stationary states on single pipes for the cases with zero slope and with nonzero slope. In the case with zero slope, the state can be represented using the Lambert--W function.
On a discrete-to-continuum convergence result for a two dimensional brittle material in the small displacement regime
Manuel Friedrich and Bernd Schmidt
2015, 10(2): 321-342 doi: 10.3934/nhm.2015.10.321 +[Abstract](359) +[PDF](499.7KB)
We consider a two-dimensional atomic mass spring system and show that in the small displacement regime the corresponding discrete energies can be related to a continuum Griffith energy functional in the sense of $\Gamma$-convergence. We also analyze the continuum problem for a rectangular bar under tensile boundary conditions and find that depending on the boundary loading the minimizers are either homogeneous elastic deformations or configurations that are completely cracked generically along a crystallographic line. As applications we discuss cleavage properties of strained crystals and an effective continuum fracture energy for magnets.
Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition
Hakima Bessaih, Yalchin Efendiev and Florin Maris
2015, 10(2): 343-367 doi: 10.3934/nhm.2015.10.343 +[Abstract](558) +[PDF](568.1KB)
The evolution Stokes equation in a domain containing periodically distributed obstacles subject to Fourier boundary condition on the boundaries is considered. We assume that the dynamic is driven by a stochastic perturbation on the interior of the domain and another stochastic perturbation on the boundaries of the obstacles. We represent the solid obstacles by holes in the fluid domain. The macroscopic (homogenized) equation is derived as another stochastic partial differential equation, defined in the whole non perforated domain. Here, the initial stochastic perturbation on the boundary becomes part of the homogenized equation as another stochastic force. We use the two-scale convergence method after extending the solution with 0 in the holes to pass to the limit. By Itô stochastic calculus, we get uniform estimates on the solution in appropriate spaces. In order to pass to the limit on the boundary integrals, we rewrite them in terms of integrals in the whole domain. In particular, for the stochastic integral on the boundary, we combine the previous idea of rewriting it on the whole domain with the assumption that the Brownian motion is of trace class. Due to the particular boundary condition dealt with, we get that the solution of the stochastic homogenized equation is not divergence free. However, it is coupled with the cell problem that has a divergence free solution. This paper represents an extension of the results of Duan and Wang (Comm. Math. Phys. 275:1508--1527, 2007), where a reaction diffusion equation with a dynamical boundary condition with a noise source term on both the interior of the domain and on the boundary was studied, and through a tightness argument and a pointwise two scale convergence method the homogenized equation was derived.
Stability of conductivities in an inverse problem in the reaction-diffusion system in electrocardiology
Bedr'Eddine Ainseba, Mostafa Bendahmane and Yuan He
2015, 10(2): 369-385 doi: 10.3934/nhm.2015.10.369 +[Abstract](437) +[PDF](462.2KB)
In this paper, we study the stability result for the conductivities diffusion coefficients to a strongly reaction-diffusion system modeling electrical activity in the heart. To study the problem, we establish a Carleman estimate for our system. The proof is based on the combination of a Carleman estimate and certain weight energy estimates for parabolic systems.
Inhomogeneities in 3 dimensional oscillatory media
Gabriela Jaramillo
2015, 10(2): 387-399 doi: 10.3934/nhm.2015.10.387 +[Abstract](327) +[PDF](409.2KB)
We consider localized perturbations to spatially homogeneous oscillations in dimension 3 using the complex Ginzburg-Landau equation as a prototype. In particular, we will focus on inhomogeneities that locally change the phase of the oscillations. In the usual translation invariant spaces and at $ \epsilon=0$ the linearization about these spatially homogeneous solutions result in an operator with zero eigenvalue embedded in the essential spectrum. In contrast, we show that when considered as an operator between Kondratiev spaces, the linearization is a Fredholm operator. These spaces consist of functions with algebraical localization that increases with each derivative. We use this result to construct solutions close to the equilibrium via the Implicit Function Theorem and derive asymptotics for wavenumbers in the far field.
Self-similar control systems and applications to zygodactyl bird's foot
Anna Chiara Lai and Paola Loreti
2015, 10(2): 401-419 doi: 10.3934/nhm.2015.10.401 +[Abstract](415) +[PDF](902.5KB)
We investigate a class of linear discrete control systems, modeling the controlled dynamics of planar manipulators as well as the skeletal dynamics of human fingers and bird's toes. A self-similarity assumption on the phalanxes allows to reinterpret the control field ruling the whole dynamics as an Iterated Function System. By exploiting this relation, we apply results coming from self-similar dynamics in order to give a geometrical description of the control system and, in particular, of its reachable set. This approach is then applied to the investigation of the zygodactyl phenomenon in birds, and in particular in parrots. This arrangement of the toes of a bird's foot, common in species living on trees, is a distribution of the foot with two toes facing forward and two back. Reachability and grasping configurations are then investigated. Finally an hybrid system modeling the owl's foot is introduced.

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