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Discrete & Continuous Dynamical Systems - S

2011 , Volume 4 , Issue 1

Issue dedicated to Claude-Michel Brauner on the occasion of his 60th birthday
Guest Editors: Jerry L. Bona, Michel Langlais and Alain Miranville

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Preface
Jerry L. Bona and Claudine Schmidt-Lainé
2011, 4(1): i-iv doi: 10.3934/dcdss.2011.4.1i +[Abstract](61) +[PDF](286.2KB)
Abstract:
This issue of Discrete and Continuous Dynamical Systems is dedicated to our friend and colleague, Claude-Michel Brauner. The papers herein are contributed by some of his admirers who come from many persuasions within the mathematical enterprise. We think this appropriate, as Claude has shown enormous breadth throughout his career. What follows is a brief appreciation of Claude-Michel, both as a scientist and as a human being.

For more information please click the “Full Text” above.
The periodic patch model for population dynamics with fractional diffusion
Henri Berestycki, Jean-Michel Roquejoffre and Luca Rossi
2011, 4(1): 1-13 doi: 10.3934/dcdss.2011.4.1 +[Abstract](154) +[PDF](180.0KB)
Abstract:
Fractional diffusions arise in the study of models from population dynamics. In this paper, we derive a class of integro-differential reaction-diffusion equations from simple principles. We then prove an approximation result for the first eigenvalue of linear integro-differential operators of the fractional diffusion type, and we study from that the dynamics of a population in a fragmented environment with fractional diffusion.
Large-time asymptotics of the generalized Benjamin-Ono-Burgers equation
Jerry L. Bona and Laihan Luo
2011, 4(1): 15-50 doi: 10.3934/dcdss.2011.4.15 +[Abstract](93) +[PDF](251.1KB)
Abstract:
In this paper, attention is given to pure initial-value problems for the generalized Benjamin-Ono-Burgers (BOB) equation

$ u_t + u_x +(P(u))_{x}-\nu $uxx$ - H$uxx=0,

where $H$ is the Hilbert transform, $\nu > 0$ and $P\ : R \to R$ is a smooth function. We study questions of global existence and of the large-time asymptotics of solutions of the initial-value problem. If $\Lambda (s)$ is defined by $\Lambda '(s) = P(s), \Lambda (0) = 0,$ then solutions of the initial-value problem corresponding to reasonable initial data maintain their integrity for all $t \geq 0$ provided that $\Lambda$ and $P'$ satisfy certain growth restrictions. In case a solution corresponding to initial data that is square integrable is global, it is straightforward to conclude it must decay to zero when $t$ becomes unboundedly large. We investigate the detailed asymptotics of this decay. For generic initial data and weak nonlinearity, it is demonstrated that the final decay is that of the linearized equation in which $P \equiv 0.$ However, if the initial data is drawn from more restricted classes that involve something akin to a condition of zero mean, then enhanced decay rates are established. These results extend the earlier work of Dix who considered the case where $P$ is a quadratic polynomial.

A doubly nonlinear parabolic equation with a singular potential
Laurence Cherfils, Stefania Gatti and Alain Miranville
2011, 4(1): 51-66 doi: 10.3934/dcdss.2011.4.51 +[Abstract](74) +[PDF](210.7KB)
Abstract:
Our aim in this paper is to study the long time behavior, in terms of finite dimensional attractors, of doubly nonlinear Allen-Cahn type equations with singular potentials.
Some remarks on the qualitative properties of solutions to a predator-prey model posed on non coincident spatial domains
Arnaud Ducrot, Vincent Guyonne and Michel Langlais
2011, 4(1): 67-82 doi: 10.3934/dcdss.2011.4.67 +[Abstract](71) +[PDF](862.1KB)
Abstract:
We are interested in the dynamical behaviour of the solution set to a two component reaction--diffusion system posed on non coincident spatial domains. The underlying biological problem is a predator--prey system featuring a non local numerical response to predation involving an integral kernel. Quite interesting while complex dynamics emerge from preliminary numerical simulations, driven both by diffusivities and by the parametric form or shape of the integral kernel. We consider a simplified version of this problem, with constant coefficients, and give some hints on the large time dynamics of solutions.
Dissipativity for a semi-linearized system modeling cellular flames
Michael Frankel, Victor Roytburd and Gregory I. Sivashinsky
2011, 4(1): 83-99 doi: 10.3934/dcdss.2011.4.83 +[Abstract](65) +[PDF](229.4KB)
Abstract:
We study a Semi-Linearized System (SLS) of second order PDEs modeling flame front dynamics. SLS is a simplified version of the weak $\kappa\theta$ model of cellular flames which is dynamically similar to the Kuramoto-Sivashinsky (KS) equation [7, 4]. We prove existence of the solutions at large, and their proximity, for finite time, to the solutions of KS. We demonstrate that SLS possesses a universal absorbing set and a compact attractor. Furthermore, we show that the attractor is of finite Hausdorff dimension.
Heteroclinic connections for multidimensional bistable reaction-diffusion equations
François Hamel and Jean-Michel Roquejoffre
2011, 4(1): 101-123 doi: 10.3934/dcdss.2011.4.101 +[Abstract](114) +[PDF](297.3KB)
Abstract:
In this paper, non-planar two-dimensional travelling fronts connecting an unstable one-dimensional periodic limiting state to a constant stable state are constructed for some reaction-diffusion equations with bistable nonlinearities. The minimal speeds are characterized in terms of the spatial period of the unstable limiting state. The limits of the minimal speeds and of the travelling fronts as the period converges to a critical minimal value or to infinity are analyzed. The fronts converge to flat fronts or to some curved fronts connecting an unstable ground state to a constant stable state.
A reaction-diffusion approximation to an area preserving mean curvature flow coupled with a bulk equation
Marie Henry, Danielle Hilhorst and Masayasu Mimura
2011, 4(1): 125-154 doi: 10.3934/dcdss.2011.4.125 +[Abstract](67) +[PDF](338.4KB)
Abstract:
Motivated by the motion of an alcohol droplet, we derive a simplified phenomenological free boundary model which consists of an area preserving mean curvature flow coupled with a bulk equation. Our aim is to introduce a nonlocal reaction-diffusion system with a small parameter $\e$ which converges to the original model as $\e$ tends to zero. This approximation enables us to overcome the technical difficulty of the free boundary problem arising in the original model.
Two-point closure based large-eddy simulations in turbulence, Part 1: Isotropic turbulence
Marcel Lesieur
2011, 4(1): 155-168 doi: 10.3934/dcdss.2011.4.155 +[Abstract](70) +[PDF](788.6KB)
Abstract:
This is the first of a series of two articles dedicated to Claude-Michel Brauner and Roger Temam on turbulence large-eddy simulations using two-point closures. The present paper deals with applications to isotropic turbulence. First, some personal memories related to my collaboration with Claude-Michel Brauner are given. Then we recall the formalism of large-eddy simulations (LES) of turbulence in physical space for flows of constant density. We consider also a passive scalar, very important for combution applications. Afterwards we study the same problem in Fourier space, on the basis of the Eddy-Damped Quasi-Normal Markovian (EDQNM) theory which is used as subgrid model. This is applied to isotropic turbulence, with particular emphasis put on turbulence decay. We discuss the issue of singularity for Euler equations. We give finally some LES results on pressure statistics.
Optimal Hölder regularity for nonautonomous Kolmogorov equations
Luca Lorenzi
2011, 4(1): 169-191 doi: 10.3934/dcdss.2011.4.169 +[Abstract](83) +[PDF](283.3KB)
Abstract:
We consider a class of nonautonomous elliptic operators A with unbounded coefficients defined in $[0,T]\times\R^N$ and we prove optimal Schauder estimates for the solution to the parabolic Cauchy problem $D_tu=$A$u+g$, $u(0,\cdot)=f$.
A mathematical model of a criminal-prone society
Juan Carlos Nuño, Miguel Angel Herrero and Mario Primicerio
2011, 4(1): 193-207 doi: 10.3934/dcdss.2011.4.193 +[Abstract](115) +[PDF](307.3KB)
Abstract:
Criminals are common to all societies. To fight against them the community takes different security measures as, for example, to bring about a police. Thus, crime causes a depletion of the common wealth not only by criminal acts but also because the cost of hiring a police force. In this paper, we present a mathematical model of a criminal-prone self-protected society that is divided into socio-economical classes. We study the effect of a non-null crime rate on a free-of-criminals society which is taken as a reference system. As a consequence, we define a criminal-prone society as one whose free-of-criminals steady state is unstable under small perturbations of a certain socio-economical context. Finally, we compare two alternative strategies to control crime: (i) enhancing police efficiency, either by enlarging its size or by updating its technology, against (ii) either reducing criminal appealing or promoting social classes at risk.
The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity
Madalina Petcu and Roger Temam
2011, 4(1): 209-222 doi: 10.3934/dcdss.2011.4.209 +[Abstract](88) +[PDF](165.8KB)
Abstract:
In the present article we consider the nonviscous Shallow Water Equations in space dimension one with Dirichlet boundary conditions for the velocity and we show the locally in time well-posedness of the model.
Best design for a fastest cells selecting process
Michel Pierre and Grégory Vial
2011, 4(1): 223-237 doi: 10.3934/dcdss.2011.4.223 +[Abstract](63) +[PDF](470.5KB)
Abstract:
We consider a cell sorting process based on negative dielectrophoresis. The goal is to optimize the shape of an electrode network to speed up the positioning. We first show that the best electric field to impose has to be radial in order to minimize the average time for a group of particles. We can get an explicit formula in the specific case of a uniform distribution of initial positions, through the resolution of the Abel integral equation. Next,we use a least-square numerical procedure to design the electrode's shape.

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