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Networks & Heterogeneous Media

2017 , Volume 12 , Issue 4

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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](63) +[HTML](23) +[PDF](525.9KB)

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](79) +[HTML](18) +[PDF](883.9KB)

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](162) +[HTML](18) +[PDF](1568.9KB)

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](73) +[HTML](13) +[PDF](547.2KB)

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](71) +[HTML](17) +[PDF](424.8KB)

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](75) +[HTML](15) +[PDF](646.0KB)

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](75) +[HTML](17) +[PDF](465.6KB)

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.

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