Coagulation, diffusion and the continuous Smoluchowski equation

The Smoluchowski equations are a system of partial differential equations modelling the diffusion and binary coagulation of a large collection of tiny particles. The mass parameter may be indexed either by positive integers or by positive reals, these corresponding to the discrete or the continuous form of the equations. For dimension d≥3, we derive the continuous Smoluchowski PDE as a kinetic limit of a microscopic model of Brownian particles liable to coalesce, using a method similar to that used to derive the discrete form of the equations in [A. Hammond, F. Rezakhanlou, The kinetic limit of a system of coagulating Brownian particles, Arch. Ration. Mech. Anal. 185 (2007) 1–67]. The principal innovation is a correlation-type bound on particle locations that permits the derivation in the continuous context while simplifying the arguments of the cited work. We also comment on the scaling satisfied by the continuous Smoluchowski PDE, and its potential implications for blow-up of solutions of the equations.     

Upscaling of a parabolic system with a large nonlinear surface reaction term

Motivated by the study of the dynamics of calcium ions in biological cells, the authors derived in [33], via periodic homogenization, a macroscopic bidomain model, by considering in the corresponding microscopic two-component problem a properly scaled nonlinear exchange term. We study here, at the microscopic scale, a similar parabolic system, with a large nonlinear interfacial reaction term. At the macroscopic scale, the nonlinear effect of this reaction term is recovered in the homogenized diffusion matrix, which is not anymore constant. This nonstandard phenomenon shows the fine interplay between reaction and diffusion in such processes.     

Homogenization of a nonlinear multiscale model of calcium dynamics in biological cells

Calcium is one of the most important intracellular messengers, which occurs in the cytosol and the endoplasmic reticulum of animal cells. While most calcium dynamics models either do not account properly for the fact that the endoplasmic reticulum constitutes a microstructure of the cell or are infeasible by resolving the fine structure very explicitly, Goel et al. [15] derived an effective macroscopic model by formal homogenization. In this paper, this approach is made rigorous using periodic homogenization techniques to upscale the nonlinear coupled system of reaction–diffusion equations and, moreover, the appropriate scaling of the interfacial exchange term is taken into consideration.     

Homogenization of cellular structures for soft tissue modelling

The aim of the paper is to show how the method of homogenization can be applied in modelling of soft tissue undergoing large deformation. Simplified microstructures are considered, which consist of hyperelastic porous matrix and periodic array of fluid‐filled cells. At the microscopic level diffusion processes are described by the Darcy law, permeability of the cellular membrane is introduced. Although at the macroscopic scale the tissue is incompressible, the flow inside microscopic volumes induces viscous relaxation effects. The homogenized problem is formulated.     

Homogenization of time harmonic Maxwell equations: the effect of interfacial currents

We consider the Maxwell equations for a composite material consisting of two phases and enjoying a periodical structure in the presence of a time‐harmonic current source. We perform the two‐scale homogenization taking into account both the interfacial layer thickness and the complex conductivity of the interfacial layer. We introduce a variational formulation of the differential system equipped with boundary and interfacial conditions. We show the unique solvability of the variational problem. Then, we analyze the low frequency case, high and very high frequency cases, with different strength of the interfacial currents. We find the macroscopic equations and determine the effective constant matrices such as the magnetic permeability, dielectric permittivity, and electric conductivity. The effective matrices depend strongly on the frequency of the current source; the dielectric permittivity and electric conductivity also depend on the strength of the interfacial currents. Copyright © 2016 John Wiley & Sons, Ltd.     

Solving Multi‐Scale problems with heterogeneous finite difference methods

Multi‐scale differential equations are problems in which the variables can have different length scales. The direct numerical solution of differential equations with multiple scales is often difficult due to the work for resolving the smallest scale. We present here a strategy which allows the use of finite difference methods for the numerical solution of parabolic multi‐scale problems, based on a coupling of macroscopic and microscopic models for the original equation.     

On Closure Conditions for the Moment Equations of a Bipolar Kinetic Model

We consider a bipolar kinetic model for charged media. In certain scalings the Debye length or the relaxation time are small. In addition different time scales are considered. These can be used in order to close the corresponding moment equations and leads to a (closed) set of macroscopic equations. We show three different scalings and obtain three completely different sets of macroscopic equations.     

The hydrodynamic limit for a system with interactions prescribed by Ginzburg-Landau type random Hamiltonian

Summary As a microscopic model we consider a system of interacting continuum like spin field overR ^d . Its evolution law is determined by the Ginzburg-Landau type random Hamiltonian and the total spin of the system is preserved by this evolution. We show that the spin field converges, under the hydrodynamic space-time scalling, to a deterministic limit which is a solution of a certain nonlinear diffusion equation. This equation describes the time evolution of the macroscopic field. The hydrodynamic scaling has an effect of the homogenization on the system at the same time.     

Homogenization of a model for water–gas flow through double‐porosity media

This paper presents a study of immiscible compressible two‐phase, such as water and gas, flow through double porosity media. The microscopic model consists of the usual equations derived from the mass conservation laws of both fluids, along with the standard Darcy–Muskat law relating the velocities to the pressure gradients and gravitational effects. The problem is written in terms of the phase formulation, that is, where the phase pressures and the phase saturations are primary unknowns. The fractured medium consists of periodically repeating homogeneous blocks and fractures, where the absolute permeability of the medium becomes discontinuous. Consequently, the model involves highly oscillatory characteristics. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. We obtain the convergence of the solutions, and a macroscopic model of the problem is constructed using the notion of two‐scale convergence combined with the dilatation technique. Copyright © 2015 John Wiley & Sons, Ltd.     

Asymptotic Analysis of the Current-Voltage Curve of a pnpn Semiconductor Device

We consider the one-dimensional steady-state semiconductor deviceequations modelling a pnpn device. There are two relevant scalingsof the equations corresponding to small and large applied voltages.In both scalings, the semiconductor equations can be consideredas singularly perturbed. It turns out that the small-voltagescaling breaks down for current values between two saturationcurrents. In that interval, the large-voltage scaling has tobe employed. For both scalings, we derive the first-order termsof an asymptotic expansion and show that the reduced problemhas a solution. An example verifies that the current-voltagecurves obtained have the expected qualitative structure.     

Homogenization and correctors for monotone problems in cylinders of small diameter

In this paper we study the homogenization of monotone diffusion equations posed in an N  -dimensional cylinder which converges to a (one-dimensional) segment line. In other terms, we pass to the limit in diffusion monotone equations posed in a cylinder whose diameter tends to zero, when simultaneously the coefficients of the equations (which are not necessarily periodic) are also varying. We obtain a limit system in both the macroscopic (one-dimensional) variable and the microscopic variable. This system is nonlocal. From this system we obtain by elimination an equation in the macroscopic variable which is local, but in contrast with usual results, the operator depends on the right-hand side of the equations. We also obtain a corrector result, i.e. an approximation of the gradients of the solutions in the strong topology of the space L^p$L^p$ in which the monotone operators are defined.     

Homogenization for contact problems with friction on rough interface

We consider a contact problem between a macroscopic solid with a smooth boundary and a technical textile, while the textile has a periodic microscopic structure and microscopically rough surface. Two–scale homogenization approach is applied to the problem. The microscopic solution is approximated in terms of macroscopic solution and some concentration factor, given as a solution of auxiliary boundary value or contact problems of elasticity on the periodicity cell. Local friction condition is represented as a continuous non–linear functional over the stress field. Two–scale convergence is used to prove the convergence of friction functional. The macroscopic initial frictional limit is found. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Droplet spreading: Intermediate scaling law by PDE methods

We consider the spreading of a thin droplet of viscous liquid on a plane surface driven by capillarity. The standard lubrication approximation leads to an evolution equation for the film height h that is ill‐posed when the spreading is limited by the no‐slip boundary condition at the liquid‐solid interface due to a singularity at the moving contact line. The most common relaxation of the no‐slip boundary condition removes this singularity but introduces a new physical length scale: the slippage length b. It is believed that this microscopic‐length scale only enters logarithmically in the effective (that is, macroscopic) spreading behavior. In this paper, we rigorously show that the naively expected spreading rate is indeed only altered by a logarithmic term involving b. More precisely, we prove a scaling law for the diameter of the apparent (that is, macroscopic) support of the droplet in time. This is an intermediate scaling law: It takes an initial layer to “forget” the initial droplet shape, whereas after a long time, the droplet is so thin that its spreading is governed by the physics on the scale b. Our proof works by deriving suitable estimates for physically relevant integral quantities: the free energy, the length of the apparent support, and their respective rates of change. As opposed to matched asymptotic methods, this PDE approach closely mimics a simple heuristic argument based on the gradient flow structure. © 2002 John Wiley & Sons, Inc.     

Global existence of a strong solution to the one-dimensional full model for irreversible phase transitions

This paper deals with the analysis of a model proposed by M. Frémond in order to describe some irreversible phase transition phenomena resulting as macroscopic effects of the microscopic movements of molecules. This model consists in a nonlinear system of partial differential equations of parabolic type and several simplifications have been studied recently. Nevertheless, up to now the question of the existence of a solution to the full problem was still open. This paper answers affirmatively to this question in the one-dimensional setting by exploiting a regularization—a priori estimates—passage to the limit procedure.     

Lagging and leading coupled continuous time random walks, renewal times and their joint limits

Subordinating a random walk to a renewal process yields a continuous time random walk (CTRW), which models diffusion and anomalous diffusion. Transition densities of scaling limits of power law CTRWs have been shown to solve fractional Fokker-Planck equations. We consider limits of CTRWs which arise when both waiting times and jumps are taken from an infinitesimal triangular array. Two different limit processes are identified when waiting times precede jumps or follow jumps, respectively, together with two limit processes corresponding to the renewal times. We calculate the joint law of all four limit processes evaluated at a fixed time t.     

Analysis of Caughley model with diffusion from mathematical ecology

We provide an analysis in function spaces of the nonlinear semigroup generated by the Caughley model with varied diffusion from mathematical ecology. The global long time asymptotic dynamics of the system of equations are well posed in the sense of an attractor. The behaviour of this attractor in small diffusion coefficients is studied. Two limit problems depending on the stability of the spatial domain in diffusion coefficients are obtained. An adequate scaling of the space variable yields a diffusion coefficients dependent spatial domain. The limit model equations are defined in the complete space of the domain and its diffusion coefficients are unitary. If the domain does not change with the diffusion coefficients, we obtain as a limit problem the system of equations with zero diffusion coefficients and no boundary conditions. The family of attractors in small diffusion coefficients is proved in the Hausdroff semidistance of sets to converge in the uniform topology of continuous functions.     

Large‐time behavior of a two‐scale semilinear reaction–diffusion system for concrete sulfatation

We study the large‐time behavior of (weak) solutions to a two‐scale reaction–diffusion system coupled with a nonlinear ordinary differential equations modeling the partly dissipative corrosion of concrete (or cement)‐based materials with sulfates. We prove that as t → ∞ , the solution to the original two‐scale system converges to the corresponding two‐scale stationary system. To obtain the main result, we make use essentially of the theory of evolution equations governed by subdifferential operators of time‐dependent convex functions developed combined with a series of two‐scale energy‐like time‐independent estimates. Copyright © 2014 John Wiley & Sons, Ltd.     

From vehicle–driver behaviors to first order traffic flow macroscopic models

This paper proposes a two scale modeling approach to vehicular traffic, where macroscopic conservation equations are closed by models at the microscopic scale obtained by a mathematical interpretation of driver behaviors to local flow conditions. The paper focuses on the closure of the mass conservation equations by phenomenological models derived by a detailed analysis at the scale of individual vehicles.