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Networks & Heterogeneous Media

June 2019 , Volume 14 , Issue 2

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Wave propagation in fractal trees. Mathematical and numerical issues
Patrick Joly, Maryna Kachanovska and Adrien Semin
2019, 14(2): 205-264 doi: 10.3934/nhm.2019010 +[Abstract](695) +[HTML](289) +[PDF](2040.29KB)
Abstract:

We propose and analyze a mathematical model for wave propagation in infinite trees with self-similar structure at infinity. This emphasis is put on the construction and approximation of transparent boundary conditions. The performance of the constructed boundary conditions is then illustrated by numerical experiments.

Derivation of second order traffic flow models with time delays
Michael Burger, Simone Göttlich and Thomas Jung
2019, 14(2): 265-288 doi: 10.3934/nhm.2019011 +[Abstract](1001) +[HTML](328) +[PDF](1337.56KB)
Abstract:

Starting from microscopic follow-the-leader models, we develop hyperbolic delay partial differential equations to govern the density and velocity of vehicular traffic. The proposed models can be seen as an extension of the classical Aw-Rascle-Zhang model, where the reaction time of drivers appears as an additional term in the velocity equation. We propose numerical methods based on first principles and present a numerical study, where we focus on the impact of time delays in comparison to undelayed models.

Newtonian fluid flow in a thin porous medium with non-homogeneous slip boundary conditions
María Anguiano and Francisco Javier Suárez-Grau
2019, 14(2): 289-316 doi: 10.3934/nhm.2019012 +[Abstract](784) +[HTML](301) +[PDF](818.83KB)
Abstract:

We consider the Stokes system in a thin porous medium \begin{document}$ \Omega_\varepsilon $\end{document} of thickness \begin{document}$ \varepsilon $\end{document} which is perforated by periodically distributed solid cylinders of size \begin{document}$ \varepsilon $\end{document}. On the boundary of the cylinders we prescribe non-homogeneous slip boundary conditions depending on a parameter \begin{document}$ \gamma $\end{document}. The aim is to give the asymptotic behavior of the velocity and the pressure of the fluid as \begin{document}$ \varepsilon $\end{document} goes to zero. Using an adaptation of the unfolding method, we give, following the values of \begin{document}$ \gamma $\end{document}, different limit systems.

A local sensitivity analysis for the kinetic Kuramoto equation with random inputs
Seung-Yeal Ha, Shi Jin and Jinwook Jung
2019, 14(2): 317-340 doi: 10.3934/nhm.2019013 +[Abstract](630) +[HTML](320) +[PDF](461.3KB)
Abstract:

We present a local sensivity analysis for the kinetic Kuramoto equation with random inputs in a large coupling regime. In our proposed random kinetic Kuramoto equation (in short, RKKE), the random inputs are encoded in the coupling strength. For the deterministic case, it is well known that the kinetic Kuramoto equation exhibits asymptotic phase concentration for well-prepared initial data in the large coupling regime. To see a response of the system to the random inputs, we provide propagation of regularity, local-in-time stability estimates for the variations of the random kinetic density function in random parameter space. For identical oscillators with the same natural frequencies, we introduce a Lyapunov functional measuring the phase concentration, and provide a local sensitivity analysis for the functional.

Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing
Mogtaba Mohammed and Mamadou Sango
2019, 14(2): 341-369 doi: 10.3934/nhm.2019014 +[Abstract](799) +[HTML](312) +[PDF](529.67KB)
Abstract:

In this paper we deal with the homogenization of stochastic nonlinear hyperbolic equations with periodically oscillating coefficients involving nonlinear damping and forcing driven by a multi-dimensional Wiener process. Using the two-scale convergence method and crucial probabilistic compactness results due to Prokhorov and Skorokhod, we show that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized problem, which is a nonlinear damped stochastic hyperbolic partial differential equation. More importantly, we also prove the convergence of the associated energies and establish a crucial corrector result.

Non-local multi-class traffic flow models
Felisia Angela Chiarello and Paola Goatin
2019, 14(2): 371-387 doi: 10.3934/nhm.2019015 +[Abstract](696) +[HTML](307) +[PDF](1158.47KB)
Abstract:

We prove the existence for small times of weak solutions for a class of non-local systems in one space dimension, arising in traffic modeling. We approximate the problem by a Godunov type numerical scheme and we provide uniform \begin{document}${{\mathbf{L}}^\infty } $\end{document} and BV estimates for the sequence of approximate solutions, locally in time. We finally present some numerical simulations illustrating the behavior of different classes of vehicles and we analyze two cost functionals measuring the dependence of congestion on traffic composition.

A model for a network of conveyor belts with discontinuous speed and capacity
Adriano Festa, Simone Göttlich and Marion Pfirsching
2019, 14(2): 389-410 doi: 10.3934/nhm.2019016 +[Abstract](657) +[HTML](297) +[PDF](1949.5KB)
Abstract:

We introduce a macroscopic model for a network of conveyor belts with various speeds and capacities. In a different way from traffic flow models, the product densities are forced to move with a constant velocity unless they reach a maximal capacity and start to queue. This kind of dynamics is governed by scalar conservation laws consisting of a discontinuous flux function. We define appropriate coupling conditions to get well-posed solutions at intersections and provide a detailed description of the solution. Some numerical simulations are presented to illustrate and confirm the theoretical results for different network configurations.

Homogenization and exact controllability for problems with imperfect interface
Sara Monsurrò and Carmen Perugia
2019, 14(2): 411-444 doi: 10.3934/nhm.2019017 +[Abstract](655) +[HTML](291) +[PDF](672.08KB)
Abstract:

The first aim of this paper is to study, by means of the periodic unfolding method, the homogenization of elliptic problems with source terms converging in a space of functions less regular than the usual \begin{document}$ L^2 $\end{document}, in an \begin{document}$ \varepsilon $\end{document}-periodic two component composite with an imperfect transmission condition on the interface. Then we exploit this result to describe the asymptotic behaviour of the exact controls and the corresponding states of hyperbolic problems set in composites with the same structure and presenting the same condition on the interface. The exact controllability is developed by applying the Hilbert Uniqueness Method, introduced by J. -L. Lions, which leads us to the construction of the exact controls as solutions of suitable transposed problem.

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