# American Institute of Mathematical Sciences

ISSN:
1556-1801

eISSN:
1556-181X

All Issues

## Networks & Heterogeneous Media

September 2018 , Volume 13 , Issue 3

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2018, 13(3): 379-407 doi: 10.3934/nhm.2018017 +[Abstract](4076) +[HTML](278) +[PDF](525.83KB)
Abstract:

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.

2018, 13(3): 409-421 doi: 10.3934/nhm.2018018 +[Abstract](5440) +[HTML](305) +[PDF](3315.05KB)
Abstract:

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.

2018, 13(3): 423-448 doi: 10.3934/nhm.2018019 +[Abstract](4534) +[HTML](210) +[PDF](717.9KB)
Abstract:

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.

2018, 13(3): 449-478 doi: 10.3934/nhm.2018020 +[Abstract](15395) +[HTML](279) +[PDF](891.74KB)
Abstract:

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.

2018, 13(3): 479-491 doi: 10.3934/nhm.2018021 +[Abstract](4783) +[HTML](203) +[PDF](389.73KB)
Abstract:

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.

2018, 13(3): 493-513 doi: 10.3934/nhm.2018022 +[Abstract](4785) +[HTML](226) +[PDF](541.19KB)
Abstract:

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.

2018, 13(3): 515-530 doi: 10.3934/nhm.2018023 +[Abstract](5955) +[HTML](254) +[PDF](494.64KB)
Abstract:

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.

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