# American Institute of Mathematical Sciences

ISSN:
1556-1801

eISSN:
1556-181X

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## Networks & Heterogeneous Media

June 2020 , Volume 15 , Issue 2

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2020, 15(2): 171-195 doi: 10.3934/nhm.2020008 +[Abstract](153) +[HTML](96) +[PDF](3565.18KB)
Abstract:

In this paper, we consider a stationary model for the flow through a network. The flow is determined by the values at the boundary nodes of the network. We call these values the loads of the network. In the applications, the feasible loads must satisfy some box constraints. We analyze the structure of the set of feasible loads. Our analysis is motivated by gas pipeline flows, where the box constraints are pressure bounds.

We present sufficient conditions that imply that the feasible set is star-shaped with respect to special points. Under stronger conditions, we prove the convexity of the set of feasible loads. All the results are given for passive networks with and without compressor stations.

This analysis is motivated by the aim to use the spheric-radial decomposition for stochastic boundary data in this model. This paper can be used for simplifying the algorithmic use of the spheric-radial decomposition.

2020, 15(2): 197-213 doi: 10.3934/nhm.2020009 +[Abstract](123) +[HTML](87) +[PDF](406.11KB)
Abstract:

In this paper we consider a family of scalar conservation laws defined on an oriented star shaped graph and we study their vanishing viscosity approximations subject to general matching conditions at the node. In particular, we prove the existence of converging subsequence and we show that the limit is a weak solution of the original problem.

2020, 15(2): 215-245 doi: 10.3934/nhm.2020010 +[Abstract](282) +[HTML](88) +[PDF](6515.01KB)
Abstract:

In this paper we propose a new mixed-primal formulation for heat-driven flows with temperature-dependent viscosity modeled by the stationary Boussinesq equations. We analyze the well-posedness of the governing equations in this mathematical structure, for which we employ the Banach fixed-point theorem and the generalized theory of saddle-point problems. The motivation is to overcome a drawback in a recent work by the authors where, in the mixed formulation for the momentum equation, the reciprocal of the viscosity is a pre-factor to a tensor product of velocities; making the analysis quite restrictive, as one needs a given continuous injection that holds only in 2D. We show in this work that by adding both the pseudo-stress and the strain rate tensors as new unknowns in the problem, we get more flexibility in the analysis, covering also the 3D case. The rest of the formulation is based on eliminating the pressure, incorporating augmented Galerkin-type terms in the mixed form of the momentum equation, and defining the normal heat flux as a suitable Lagrange multiplier in a primal formulation for the energy equation. Additionally, the symmetry of the stress is imposed in an ultra-weak sense, and consequently the vorticity tensor is no longer required as part of the unknowns. A finite element method that follows the same setting is then proposed, where we remark that both pressure and vorticity can be recovered from the principal unknowns via postprocessing formulae. The solvability of the discrete problem is analyzed by means of the Brouwer fixed-point theorem, and we derive error estimates in suitable norms. Numerical examples illustrate the performance of the new schem and its use in the simulation of mantle convection, and they also confirm the theoretical rates of convergence.

2020, 15(2): 247-259 doi: 10.3934/nhm.2020011 +[Abstract](137) +[HTML](50) +[PDF](5577.21KB)
Abstract:

In this paper, we construct approximated solutions of Differential Equations (DEs) using the Deep Neural Network (DNN). Furthermore, we present an architecture that includes the process of finding model parameters through experimental data, the inverse problem. That is, we provide a unified framework of DNN architecture that approximates an analytic solution and its model parameters simultaneously. The architecture consists of a feed forward DNN with non-linear activation functions depending on DEs, automatic differentiation [2], reduction of order, and gradient based optimization method. We also prove theoretically that the proposed DNN solution converges to an analytic solution in a suitable function space for fundamental DEs. Finally, we perform numerical experiments to validate the robustness of our simplistic DNN architecture for 1D transport equation, 2D heat equation, 2D wave equation, and the Lotka-Volterra system.

2020, 15(2): 261-279 doi: 10.3934/nhm.2020012 +[Abstract](143) +[HTML](52) +[PDF](482.3KB)
Abstract:

We qualitatively compare the solutions of a multilane model with those produced by the classical Lighthill-Whitham-Richards equation with suitable coupling conditions at simple road junctions. The numerical simulations are based on the Godunov and upwind schemes. Several tests illustrate the models' behaviour in different realistic situations.

2020, 15(2): 281-305 doi: 10.3934/nhm.2020013 +[Abstract](65) +[HTML](29) +[PDF](413.96KB)
Abstract:

We consider the Dirichlet problem for an elliptic multivalued maximal monotone operator \begin{document}${\mathcal A}_\varepsilon$\end{document} satisfying growth estimates of power type with a variable exponent. This exponent \begin{document}$p_\varepsilon(x)$\end{document} and also the symbol of the operator \begin{document}${\mathcal A}_\varepsilon$\end{document} oscillate with a small period \begin{document}$\varepsilon$\end{document} with respect to the space variable \begin{document}$x$\end{document}. We prove a homogenization result for this problem.

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