# American Institute of Mathematical Sciences

ISSN:
1556-1801

eISSN:
1556-181X

All Issues

## Networks & Heterogeneous Media

December 2019 , Volume 14 , Issue 4

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2019, 14(4): 633-657 doi: 10.3934/nhm.2019025 +[Abstract](1799) +[HTML](185) +[PDF](391.68KB)
Abstract:

This paper is concerned with a target-detection model using bio-nanomachines in the human body that is actively being discussed in the field of molecular communication networks. Although the model was originally proposed as spatially one-dimensional, here we extend it to two dimensions and analyze it. After the mathematical formulation, we first verify the solvability of the stationary problem, and then the existence of a strong global-in-time solution of the non-stationary problem in Sobolev–Slobodetskiĭ space. We also show the non-negativeness of the non-stationary solution.

2019, 14(4): 659-676 doi: 10.3934/nhm.2019026 +[Abstract](2194) +[HTML](179) +[PDF](1618.77KB)
Abstract:

Gas flow through pipeline networks can be described using \begin{document}$2\times 2$\end{document} hyperbolic balance laws along with coupling conditions at nodes. The numerical solution at steady state is highly sensitive to these coupling conditions and also to the balance between flux and source terms within the pipes. To avoid spurious oscillations for near equilibrium flows, it is essential to design well-balanced schemes. Recently Chertock, Herty & Özcan[11] introduced a well-balanced method for general \begin{document}$2\times 2$\end{document} systems of balance laws. In this paper, we simplify and extend this approach to a network of pipes. We prove well-balancing for different coupling conditions and for compressors stations, and demonstrate the advantage of the scheme by numerical experiments.

2019, 14(4): 677-707 doi: 10.3934/nhm.2019027 +[Abstract](1754) +[HTML](149) +[PDF](473.41KB)
Abstract:

We study the Vlasov-Poisson-Fokker-Planck (VPFP) system with uncertainty and multiple scales. Here the uncertainty, modeled by multi-dimensional random variables, enters the system through the initial data, while the multiple scales lead the system to its high-field or parabolic regimes. We obtain a sharp decay rate of the solution to the global Maxwellian, which reveals that the VPFP system is decreasingly sensitive to the initial perturbation as the Knudsen number goes to zero. The sharp regularity estimates in terms of the Knudsen number lead to the stability of the generalized Polynomial Chaos stochastic Galerkin (gPC-SG) method. Based on the smoothness of the solution in the random space and the stability of the numerical method, we conclude the gPC-SG method has spectral accuracy uniform in the Knudsen number.

2019, 14(4): 709-732 doi: 10.3934/nhm.2019028 +[Abstract](1561) +[HTML](196) +[PDF](584.2KB)
Abstract:

We consider two scalar conservation laws with non-local flux functions, describing traffic flow on roads with rough conditions. In the first model, the velocity of the car depends on an averaged downstream density, while in the second model one considers an averaged downstream velocity. The road condition is piecewise constant with a jump at \begin{document}$x = 0$\end{document}. We study stationary traveling wave profiles cross \begin{document}$x = 0$\end{document}, for all possible cases. We show that, depending on the case, there could exit infinitely many profiles, a unique profile, or no profiles at all. Furthermore, some of the profiles are time asymptotic solutions for the Cauchy problem of the conservation laws under mild assumption on the initial data, while other profiles are unstable.

2019, 14(4): 733-757 doi: 10.3934/nhm.2019029 +[Abstract](2044) +[HTML](152) +[PDF](545.92KB)
Abstract:

We address a spectral problem for the Dirichlet-Laplace operator in a waveguide \begin{document}$\Pi^ \varepsilon$\end{document}. \begin{document}$\Pi^ \varepsilon$\end{document} is obtained from repsilon an unbounded two-dimensional strip \begin{document}$\Pi$\end{document} which is periodically perforated by a family of holes, which are also periodically distributed along a line, the so-called "perforation string". We assume that the two periods are different, namely, \begin{document}$O(1)$\end{document} and \begin{document}$O( \varepsilon)$\end{document} respectively, where \begin{document}$0< \varepsilon\ll 1$\end{document}. We look at the band-gap structure of the spectrum \begin{document}$\sigma^ \varepsilon$\end{document} as \begin{document}$\varepsilon\to 0$\end{document}. We derive asymptotic formulas for the endpoints of the spectral bands and show that \begin{document}$\sigma^ \varepsilon$\end{document} has a large number of short bands of length \begin{document}$O( \varepsilon)$\end{document} which alternate with wide gaps of width \begin{document}$O(1)$\end{document}.

2019, 14(4): 759-769 doi: 10.3934/nhm.2019030 +[Abstract](1313) +[HTML](161) +[PDF](310.28KB)
Abstract:

We study the Schrödinger-Lohe model. Making use of the principal fundamental matrix \begin{document}$Y$\end{document} of linear ODEs with variable coefficients, the coupled nonlinear Schrödinger-Lohe system is transformed into the decoupled linear Schrödinger equations. The boundedness of \begin{document}$Y$\end{document} is shown for the case of complete synchronization. We also study the cases where the principal fundamental matrices can be derived explicitly.

2019, 14(4): 771-788 doi: 10.3934/nhm.2019031 +[Abstract](1784) +[HTML](141) +[PDF](360.52KB)
Abstract:

The outcome of elections is strongly dependent on the districting choices, making thus possible (and frequent) the gerrymandering phenomenon, i.e. politicians suitably changing the shape of electoral districts in order to win the forthcoming elections. While so far the problem has been treated using continuous analysis tools, it has been recently pointed out that a more reality-adherent model would use the discrete geometry of graphs or networks. Here we propose a parameter-dependent discrete model for choosing an "optimal" districting plan. We analyze several properties of the model and lay foundations for further analysis on the subject.

2019, 14(4): 789-804 doi: 10.3934/nhm.2019032 +[Abstract](1725) +[HTML](161) +[PDF](355.43KB)
Abstract:

We study a Cucker-Smale-type flocking model with distributed time delay where individuals interact with each other through normalized communication weights. Based on a Lyapunov functional approach, we provide sufficient conditions for the velocity alignment behavior. We then show that as the number of individuals \begin{document}$N$\end{document} tends to infinity, the \begin{document}$N$\end{document}-particle system can be well approximated by a delayed Vlasov alignment equation. Furthermore, we also establish the global existence of measure-valued solutions for the delayed Vlasov alignment equation and its large-time asymptotic behavior.

2020 Impact Factor: 1.213
5 Year Impact Factor: 1.384
2020 CiteScore: 1.9