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Volume 13, 2018

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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.

Note: “Most Cited” is by Cross-Ref , and “Most Downloaded” is based on available data in the new website.

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Propagation of regularity and finite-time collisions for the thermomechanical Cucker-Smale model with a singular communication
Young-Pil Choi, Seung-Yeal Ha and Jeongho Kim
2018, 13(3) : 379-407 doi: 10.3934/nhm.2018017 +[Abstract](88) +[HTML](20) +[PDF](525.83KB)

We study dynamical behaviors of the ensemble of thermomechanical Cucker-Smale (in short TCS) particles with singular power-law communication weights in velocity and temperatures. For the particle TCS model, we present several sufficient frameworks for the global regularity of solution and a finite-time breakdown depending on the blow-up exponents in the power-law communication weights at the origin where the relative spatial distances become zero. More precisely, when the blow-up exponent in velocity communication weight is greater than unity and the blow-up exponent in temperature communication weights is more than twice of blow-up exponent in velocity communication, we show that there will be no finite time collision between particles, unless there are collisions initially. In contrast, when the blow-up exponent of velocity communication weight is smaller than unity, we show that there can be a collision in finite time. For the kinetic TCS equation, we present a local-in-time existence of a unique weak solution using the suitable regularization and compactness arguments.

Follow-the-Leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow
Helge Holden and Nils Henrik Risebro
2018, 13(3) : 409-421 doi: 10.3934/nhm.2018018 +[Abstract](54) +[HTML](19) +[PDF](3315.05KB)

We show how to view the standard Follow-the-Leader (FtL) model as a numerical method to compute numerically the solution of the Lighthill-Whitham-Richards (LWR) model for traffic flow. As a result we offer a simple proof that FtL models converge to the LWR model for traffic flow when traffic becomes dense. The proof is based on techniques used in the analysis of numerical schemes for conservation laws, and the equivalence of weak entropy solutions of conservation laws in the Lagrangian and Eulerian formulation.

Perturbations of minimizing movements and curves of maximal slope
Antonio Tribuzio
2018, 13(3) : 423-448 doi: 10.3934/nhm.2018019 +[Abstract](54) +[HTML](19) +[PDF](717.9KB)

We modify the De Giorgi's minimizing movements scheme for a functional $φ$, by perturbing the dissipation term, and find a condition on the perturbations which ensures the convergence of the scheme to an absolutely continuous perturbed minimizing movement. The perturbations produce a variation of the metric derivative of the minimizing movement. This process is formalized by the introduction of the notion of curve of maximal slope for $φ$ with a given rate. We show that if we relax the condition on the perturbations we may have many different meaningful effects; in particular, some perturbed minimizing movements may explore different potential wells.

Traveling wave profiles for a Follow-the-Leader model for traffic flow with rough road condition
Wen Shen
2018, 13(3) : 449-478 doi: 10.3934/nhm.2018020 +[Abstract](55) +[HTML](20) +[PDF](891.74KB)

We study a Follow-the-Leader (FtL) ODE model for traffic flow with rough road condition, and analyze stationary traveling wave profiles where the solutions of the FtL model trace along, near the jump in the road condition. We derive a discontinuous delay differential equation (DDDE) for these profiles. For various cases, we obtain results on existence, uniqueness and local stability of the profiles. The results here offer an alternative approximation, possibly more realistic than the classical vanishing viscosity approach, to the conservation law with discontinuous flux for traffic flow.

Interior regularity to the steady incompressible shear thinning fluids with non-Standard growth
Hyeong-Ohk Bae, Hyoungsuk So and Yeonghun Youn
2018, 13(3) : 479-491 doi: 10.3934/nhm.2018021 +[Abstract](42) +[HTML](15) +[PDF](389.73KB)

We consider weak solutions to the equations of stationary motion of a class of non-Newtonian fluids which includes the power law model. The power depends on the spatial variable, which is motivated by electrorheological fluids. We prove the existence of second order derivatives of weak solutions in the shear thinning cases.

Crystalline evolutions in chessboard-like microstructures
Annalisa Malusa and Matteo Novaga
2018, 13(3) : 493-513 doi: 10.3934/nhm.2018022 +[Abstract](49) +[HTML](20) +[PDF](541.19KB)

We describe the macroscopic behavior of evolutions by crystalline curvature of planar sets in a chessboard-like medium, modeled by a periodic forcing term. We show that the underlying microstructure may produce both pinning and confinement effects on the geometric motion.

SIR Rumor spreading model with trust rate distribution
Bum Il Hong, Nahmwoo Hahm and Sun-Ho Choi
2018, 13(3) : 515-530 doi: 10.3934/nhm.2018023 +[Abstract](77) +[HTML](30) +[PDF](494.64KB)

In this paper, we study a rumor spreading model in which several types of ignorants exist with trust rate distributions \begin{document}$λ_i $\end{document}, \begin{document}$≤ i≤ N$\end{document}. We rigorously show the existence of a threshold on a momentum type initial quantity related to rumor outbreak occurrence regardless of the total initial population. We employ a steady state analysis to obtain the final size of the rumor. Using numerical simulations, we demonstrate the analytical result in which the threshold phenomenon exists for rumor size and discuss interaction between the ignorants of several types of trust rates.

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](783) +[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](600) +[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](903) +[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](654) +[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](775) +[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](655) +[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](582) +[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](557) +[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](685) +[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](482) +[PDF](212.0KB) Cited By(30)
Functional model for extensions of symmetric operators and applications to scattering theory
Kirill D. Cherednichenko, Alexander V. Kiselev and Luis O. Silva
2018, 13(2) : 191-215 doi: 10.3934/nhm.2018009 +[Abstract](306) +[HTML](151) +[PDF](571.43KB) PDF Downloads(75)
A two-dimensional data-driven model for traffic flow on highways
Michael Herty, Adrian Fazekas and Giuseppe Visconti
2018, 13(2) : 217-240 doi: 10.3934/nhm.2018010 +[Abstract](312) +[HTML](185) +[PDF](7915.97KB) PDF Downloads(65)
Uniform stability and mean-field limit for the augmented Kuramoto model
Seung-Yeal Ha, Jeongho Kim, Jinyeong Park and Xiongtao Zhang
2018, 13(2) : 297-322 doi: 10.3934/nhm.2018013 +[Abstract](267) +[HTML](123) +[PDF](494.85KB) PDF Downloads(64)
On a vorticity-based formulation for reaction-diffusion-Brinkman systems
Verónica Anaya, Mostafa Bendahmane, David Mora and Ricardo Ruiz Baier
2018, 13(1) : 69-94 doi: 10.3934/nhm.2018004 +[Abstract](448) +[HTML](324) +[PDF](6157.76KB) PDF Downloads(63)
Derivation of a rod theory from lattice systems with interactions beyond nearest neighbours
Roberto Alicandro, Giuliano Lazzaroni and Mariapia Palombaro
2018, 13(1) : 1-26 doi: 10.3934/nhm.2018001 +[Abstract](414) +[HTML](196) +[PDF](360.33KB) PDF Downloads(57)
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](736) +[PDF](2266.2KB) PDF Downloads(56)
Stationary solutions and asymptotic behaviour for a chemotaxis hyperbolic model on a network
Francesca R. Guarguaglini
2018, 13(1) : 47-67 doi: 10.3934/nhm.2018003 +[Abstract](369) +[HTML](203) +[PDF](524.21KB) PDF Downloads(43)
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](609) +[HTML](203) +[PDF](525.9KB) PDF Downloads(40)
Formation, stability and basin of phase-locking for Kuramoto oscillators bidirectionally coupled in a ring
Xiaoxue Zhao, Zhuchun Li and Xiaoping Xue
2018, 13(2) : 323-337 doi: 10.3934/nhm.2018014 +[Abstract](168) +[HTML](147) +[PDF](417.49KB) PDF Downloads(38)
Stability and implementation of a cycle-based max pressure controller for signalized traffic networks
Leah Anderson, Thomas Pumir, Dimitrios Triantafyllos and Alexandre M. Bayen
2018, 13(2) : 241-260 doi: 10.3934/nhm.2018011 +[Abstract](243) +[HTML](142) +[PDF](936.84KB) PDF Downloads(37)

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