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eISSN:
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Volume 12, 2017

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Volume 2, 2007

Volume 1, 2006

NHM offers a strong combination of three features: Interdisciplinary character, specific focus, and deep mathematical content. Also, the journal aims to create a link between the discrete and the continuous communities, which distinguishes it from other journals with strong PDE orientation.

NHM publishes original contributions of high quality in networks, heterogeneous media and related fields. NHM is thus devoted to research work on complex media arising in mathematical, physical, engineering, socio-economical and bio-medical problems.

  • AIMS is a member of COPE. All AIMS journals adhere to the publication ethics and malpractice policies outlined by COPE.
  • Publishes 4 issues a year in March, June, September and December.
  • Publishes online only.
  • Indexed in Science Citation Index-Expanded, ISI Alerting Services, CompuMath Citation Index, Current Contents/Physical, Chemical & Earth Sciences (CC/PC&ES), INSPEC, Mathematical Reviews, MathSciNet, PASCAL/CNRS, Scopus, Web of Science and Zentralblatt MATH.
  • Archived in Portico and CLOCKSS.
  • NHM is a publication of the American Institute of Mathematical Sciences. All rights reserved.

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Homogenization of Stokes system using Bloch waves
Grégoire Allaire  , Tuhin Ghosh  and  Muthusamy Vanninathan 
2017, 12(4) : 525-550 doi: 10.3934/nhm.2017022 +[Abstract](51) +[HTML](20) +[PDF](525.9KB)
Abstract:

In this work, we study the Bloch wave homogenization for the Stokes system with periodic viscosity coefficient. In particular, we obtain the spectral interpretation of the homogenized tensor. The presence of the incompressibility constraint in the model raises new issues linking the homogenized tensor and the Bloch spectral data. The main difficulty is a lack of smoothness for the bottom of the Bloch spectrum, a phenomenon which is not present in the case of the elasticity system. This issue is solved in the present work, completing the homogenization process of the Stokes system via the Bloch wave method.

Sharp interface limit in a phase field model of cell motility
Leonid Berlyand  , Mykhailo Potomkin  and  Volodymyr Rybalko 
2017, 12(4) : 551-590 doi: 10.3934/nhm.2017023 +[Abstract](67) +[HTML](12) +[PDF](883.9KB)
Abstract:

We consider a phase field model of cell motility introduced in [40] which consists of two coupled parabolic PDEs. We study the asymptotic behavior of solutions in the limit of a small parameter related to the width of the interface (sharp interface limit). We formally derive an equation of motion of the interface, which is mean curvature motion with an additional nonlinear term. In a 1D model parabolic problem we rigorously justify the sharp interface limit. To this end, a special representation of solutions is introduced, which reduces analysis of the system to a single nonlinear PDE that describes the interface velocity. Further stability analysis reveals a qualitative change in the behavior of the system for small and large values of the coupling parameter. Using numerical simulations we also show discontinuities of the interface velocity and hysteresis. Also, in the 1D case we establish nontrivial traveling waves when the coupling parameter is large enough.

Asymptotic problems and numerical schemes for traffic flows with unilateral constraints describing the formation of jams
Florent Berthelin  , Thierry Goudon  , Bastien Polizzi  and  Magali Ribot 
2017, 12(4) : 591-617 doi: 10.3934/nhm.2017024 +[Abstract](147) +[HTML](13) +[PDF](1568.9KB)
Abstract:

We discuss numerical strategies to deal with PDE systems describing traffic flows, taking into account a density threshold, which restricts the vehicle density in the situation of congestion. These models are obtained through asymptotic arguments. Hence, we are interested in the simulation of approached models that contain stiff terms and large speeds of propagation. We design schemes intended to apply with relaxed stability conditions.

A multiscale model reduction method for nonlinear monotone elliptic equations in heterogeneous media
Eric Chung  , Yalchin Efendiev  , Ke Shi  and  Shuai Ye 
2017, 12(4) : 619-642 doi: 10.3934/nhm.2017025 +[Abstract](59) +[HTML](11) +[PDF](547.2KB)
Abstract:

In this paper, we present a multiscale model reduction framework within Generalized Multiscale Finite Element Method (GMsFEM) for nonlinear elliptic problems. We consider an exemplary problem, which consists of nonlinear p-Laplacian with heterogeneous coefficients. The main challenging feature of this problem is that local subgrid models are nonlinear involving the gradient of the solution (e.g., in the case of scale separation, when using homogenization). Our main objective is to develop snapshots and local spectral problems, which are the main ingredients of GMsFEM, for these problems. Our contributions can be summarized as follows. (1) We re-cast the multiscale model reduction problem onto the boundaries of coarse cells. This is important and allows capturing separable scales as discussed. (2) We introduce nonlinear eigenvalue problems in the snapshot space for these nonlinear "harmonic" functions. (3) We present convergence analysis and numerical results, which show that our approaches can recover the fine-scale solution with a few degrees of freedom. The proposed methods can, in general, be used for more general nonlinear problems, where one needs nonlinear local spectral decomposition.

The Lax-Oleinik semigroup on graphs
Renato Iturriaga  and  Héctor Sánchez Morgado 
2017, 12(4) : 643-662 doi: 10.3934/nhm.2017026 +[Abstract](55) +[HTML](13) +[PDF](424.8KB)
Abstract:

We consider Tonelli Lagrangians on a graph, define weak KAM solutions, which happen to be the fixed points of the Lax-Oleinik semi-group, and identify their uniqueness set as the Aubry set, giving a representation formula. Our main result is the long time convergence of the Lax Oleinik semi-group. It follows that weak KAM solutions are viscosity solutions of the Hamilton-Jacobi equation [3, 4], and in the case of Hamiltonians called of eikonal type in [3], we prove that the converse holds.

Capacity drop and traffic control for a second order traffic model
Oliver Kolb  , Simone Göttlich  and  Paola Goatin 
2017, 12(4) : 663-681 doi: 10.3934/nhm.2017027 +[Abstract](61) +[HTML](12) +[PDF](646.0KB)
Abstract:

In this paper, we illustrate how second order traffic flow models, in our case the Aw-Rascle equations, can be used to reproduce empirical observations such as the capacity drop at merges and solve related optimal control problems. To this aim, we propose a model for on-ramp junctions and derive suitable coupling conditions. These are associated to the first order Godunov scheme to numerically study the well-known capacity drop effect, where the outflow of the system is significantly below the expected maximum. Control issues such as speed and ramp meter control are also addressed in a first-discretize-then-optimize framework.

Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces
Ioannis Markou 
2017, 12(4) : 683-705 doi: 10.3934/nhm.2017028 +[Abstract](60) +[HTML](14) +[PDF](465.6KB)
Abstract:

In this paper we study the hydrodynamic (small mass approximation) limit of a Fokker-Planck equation. This equation arises in the kinetic description of the evolution of a particle system immersed in a viscous Stokes flow. We discuss two different methods of hydrodynamic convergence. The first method works with initial data in a weighted L2 space and uses weak convergence and the extraction of convergent subsequences. The second uses entropic initial data and gives an L1 convergence to the solution of the limit problem via the study of the relative entropy.

Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity
M.A.J Chaplain  and  G. Lolas 
2006, 1(3) : 399-439 doi: 10.3934/nhm.2006.1.399 +[Abstract](126) +[PDF](993.7KB) Cited By(67)
Gas flow in pipeline networks
Mapundi K. Banda  , Michael Herty  and  Axel Klar 
2006, 1(1) : 41-56 doi: 10.3934/nhm.2006.1.41 +[Abstract](62) +[PDF](3881.2KB) Cited By(64)
Distributed model predictive control of irrigation canals
Rudy R. Negenborn  , Peter-Jules van Overloop  , Tamás Keviczky  and  Bart De Schutter 
2009, 4(2) : 359-380 doi: 10.3934/nhm.2009.4.359 +[Abstract](181) +[PDF](281.3KB) Cited By(63)
Spreading speed revisited: Analysis of a free boundary model
Gary Bunting  , Yihong Du  and  Krzysztof Krakowski 
2012, 7(4) : 583-603 doi: 10.3934/nhm.2012.7.583 +[Abstract](58) +[PDF](737.9KB) Cited By(56)
On the variational theory of traffic flow: well-posedness, duality and applications
Carlos F. Daganzo 
2006, 1(4) : 601-619 doi: 10.3934/nhm.2006.1.601 +[Abstract](109) +[PDF](337.5KB) Cited By(53)
K-core decomposition of Internet graphs: hierarchies, self-similarity and measurement biases
José Ignacio Alvarez-Hamelin  , Luca Dall'Asta  , Alain Barrat  and  Alessandro Vespignani 
2008, 3(2) : 371-393 doi: 10.3934/nhm.2008.3.371 +[Abstract](70) +[PDF](739.3KB) Cited By(52)
Coupling conditions for gas networks governed by the isothermal Euler equations
Mapundi K. Banda  , Michael Herty  and  Axel Klar 
2006, 1(2) : 295-314 doi: 10.3934/nhm.2006.1.295 +[Abstract](104) +[PDF](1720.1KB) Cited By(48)
Large time behavior of nonlocal aggregation models with nonlinear diffusion
Martin Burger  and  Marco Di Francesco 
2008, 3(4) : 749-785 doi: 10.3934/nhm.2008.3.749 +[Abstract](63) +[PDF](412.0KB) Cited By(45)
Explicit solutions of some linear-quadratic mean field games
Martino Bardi 
2012, 7(2) : 243-261 doi: 10.3934/nhm.2012.7.243 +[Abstract](49) +[PDF](433.5KB) Cited By(38)
A Well Posed Riemann Problem for the $p$--System at a Junction
Rinaldo M. Colombo  and  Mauro Garavello 
2006, 1(3) : 495-511 doi: 10.3934/nhm.2006.1.495 +[Abstract](42) +[PDF](212.0KB) Cited By(30)
Sparse stabilization of dynamical systems driven by attraction and avoidance forces
Mattia Bongini  and  Massimo Fornasier 
2014, 9(1) : 1-31 doi: 10.3934/nhm.2014.9.1 +[Abstract](78) +[PDF](2266.2KB) PDF Downloads(52)
Numerical network models and entropy principles for isothermal junction flow
Gunhild A. Reigstad 
2014, 9(1) : 65-95 doi: 10.3934/nhm.2014.9.65 +[Abstract](68) +[PDF](1803.6KB) PDF Downloads(5)
Numerical approximation of a coagulation-fragmentation model for animal group size statistics
Pierre Degond  and  Maximilian Engel 
2017, 12(2) : 217-243 doi: 10.3934/nhm.2017009 +[Abstract](66) +[HTML](23) +[PDF](819.2KB) PDF Downloads(4)
Force-based models of pedestrian dynamics
Mohcine Chraibi  , Ulrich Kemloh  , Andreas Schadschneider  and  Armin Seyfried 
2011, 6(3) : 425-442 doi: 10.3934/nhm.2011.6.425 +[Abstract](101) +[PDF](729.9KB) PDF Downloads(3)
A discrete Hughes model for pedestrian flow on graphs
Fabio Camilli  , Adriano Festa  and  Silvia Tozza 
2017, 12(1) : 93-112 doi: 10.3934/nhm.2017004 +[Abstract](97) +[HTML](0) +[PDF](1093.6KB) PDF Downloads(2)
Gas flow in pipeline networks
Mapundi K. Banda  , Michael Herty  and  Axel Klar 
2006, 1(1) : 41-56 doi: 10.3934/nhm.2006.1.41 +[Abstract](62) +[PDF](3881.2KB) PDF Downloads(2)
Control of systems of conservation laws with boundary errors
Christophe Prieur 
2009, 4(2) : 393-407 doi: 10.3934/nhm.2009.4.393 +[Abstract](154) +[PDF](295.9KB) PDF Downloads(2)
On the modeling of crowd dynamics: Looking at the beautiful shapes of swarms
Nicola Bellomo  and  Abdelghani Bellouquid 
2011, 6(3) : 383-399 doi: 10.3934/nhm.2011.6.383 +[Abstract](110) +[PDF](418.5KB) PDF Downloads(2)
Constructing set-valued fundamental diagrams from Jamiton solutions in second order traffic models
Benjamin Seibold  , Morris R. Flynn  , Aslan R. Kasimov  and  Rodolfo R. Rosales 
2013, 8(3) : 745-772 doi: 10.3934/nhm.2013.8.745 +[Abstract](89) +[PDF](1439.2KB) PDF Downloads(2)
Fluid-structure interaction in a pre-stressed tube with thick elastic walls I: the stationary Stokes problem
Andro Mikelić  , Giovanna Guidoboni  and  Sunčica Čanić 
2007, 2(3) : 397-423 doi: 10.3934/nhm.2007.2.397 +[Abstract](35) +[PDF](344.6KB) PDF Downloads(2)

2016  Impact Factor: 1.2

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