# American Institute of Mathematical Sciences

ISSN:
1556-1801

eISSN:
1556-181X

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## Networks and Heterogeneous Media

December 2020 , Volume 15 , Issue 4

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2020, 15(4): 555-580 doi: 10.3934/nhm.2020014 +[Abstract](1207) +[HTML](256) +[PDF](666.72KB)
Abstract:

We consider the spectrum of a class of positive, second-order elliptic systems of partial differential equations defined in the plane \begin{document}$\mathbb{R}^2$\end{document}. The coefficients of the equation are assumed to have a special form, namely, they are doubly periodic and of high contrast. More precisely, the plane \begin{document}$\mathbb{R}^2$\end{document} is decomposed into an infinite union of the translates of the rectangular periodicity cell \begin{document}$\Omega^0$\end{document}, and this in turn is divided into two components, on each of which the coefficients have different, constant values. Moreover, the second component of \begin{document}$\Omega^0$\end{document} consist of a neighborhood of the boundary of the cell of the width \begin{document}$h$\end{document} and thus has an area comparable to \begin{document}$h$\end{document}, where \begin{document}$h>0$\end{document} is a small parameter.

Using the methods of asymptotic analysis we study the position of the spectral bands as \begin{document}$h \to 0$\end{document} and in particular show that the spectrum has at least a given, arbitrarily large number of gaps, provided \begin{document}$h$\end{document} is small enough.

2020, 15(4): 581-603 doi: 10.3934/nhm.2020015 +[Abstract](1328) +[HTML](264) +[PDF](597.34KB)
Abstract:

We consider the effective conductivity \begin{document}$\lambda^{\mathrm{eff}}$\end{document} of a periodic two-phase composite obtained by introducing into an infinite homogeneous matrix a periodic set of inclusions of a different material. Then we study the behavior of \begin{document}$\lambda^{\mathrm{eff}}$\end{document} upon perturbation of the shape of the inclusions, of the periodicity structure, and of the conductivity of each material.

2020, 15(4): 605-631 doi: 10.3934/nhm.2020016 +[Abstract](1515) +[HTML](279) +[PDF](753.87KB)
Abstract:

We introduce new coupling conditions for isentropic flow on networks based on an artificial density at the junction. The new coupling conditions can be derived from a kinetic model by imposing a condition on energy dissipation. Existence and uniqueness of solutions to the generalized Riemann and Cauchy problem are proven. The result for the generalized Riemann problem is globally in state space. Furthermore, non-increasing energy at the junction and a maximum principle are proven. A numerical example is given in which the new conditions are the only known conditions leading to the physically correct wave types. The approach generalizes to full gas dynamics.

2020, 15(4): 633-652 doi: 10.3934/nhm.2020017 +[Abstract](1249) +[HTML](215) +[PDF](845.87KB)
Abstract:

We investigate the simultaneous observability of infinite systems of vibrating strings or beams having a common endpoint where the observation is taking place. Our results are new even for finite systems because we allow the vibrations to take place in independent directions. Our main tool is a vectorial generalization of some classical theorems of Ingham, Beurling and Kahane in nonharmonic analysis.

2020, 15(4): 653-679 doi: 10.3934/nhm.2020018 +[Abstract](1805) +[HTML](254) +[PDF](1245.47KB)
Abstract:

We study the modeling and simulation of gas pipeline networks, with a focus on fast numerical methods for the simulation of transient dynamics. The obtained mathematical model of the underlying network is represented by a system of nonlinear differential algebraic equations (DAEs). With our modeling approach, we reduce the number of algebraic constraints, which correspond to the \begin{document}$(2,2)$\end{document} block in our semi-explicit DAE model, to the order of junction nodes in the network, where a junction node couples at least three pipelines. We can furthermore ensure that the \begin{document}$(1, 1)$\end{document} block of all system matrices including the Jacobian is block lower triangular by using a specific ordering of the pipes of the network. We then exploit this structure to propose an efficient preconditioner for the fast simulation of the network. We test our numerical methods on benchmark problems of (well-)known gas networks and the numerical results show the efficiency of our methods.

2020, 15(4): 681-710 doi: 10.3934/nhm.2020019 +[Abstract](1253) +[HTML](207) +[PDF](660.24KB)
Abstract:

Here, we study the existence and the convergence of solutions for the vanishing discount MFG problem with a quadratic Hamiltonian. We give conditions under which the discounted problem has a unique classical solution and prove convergence of the vanishing-discount limit to a unique solution up to constants. Then, we establish refined asymptotics for the limit. When those conditions do not hold, the limit problem may not have a unique solution and its solutions may not be smooth, as we illustrate in an elementary example. Finally, we investigate the stability of regular weak solutions and address the selection problem. Using ideas from Aubry-Mather theory, we establish a selection criterion for the limit.

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