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Discrete & Continuous Dynamical Systems - B

2008 , Volume 10 , Issue 1

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Estimates of the wake for the 3D Oseen equations
Hyeong-Ohk Bae and  Bum Ja Jin
2008, 10(1): 1-18 doi: 10.3934/dcdsb.2008.10.1 +[Abstract](50) +[PDF](229.7KB)
Since there is a wake for the Oseen flow, the convergence rates inside and outside the wake region might be different. The existence of the wake for the steady Oseen flow is given in Galdi [17]. We estimate these rates for the non-steady case. In order to do that, we estimate the $L^p$ spatial-temporal decay rates of solutions of incompressible Oseen flow in an exterior domain. For the spatial-temporal decays, we consider weights of the form $|x|$ and $|x-\mathbf u_\infty t|$, where $x$ is the spatial variable and $\mathbf u_\infty$ is the constant velocity at infinity. From our estimates, we conclude the convergence inside the wake region might be slower than outside the wake region.
Shadowing for discrete approximations of abstract parabolic equations
Wolf-Jürgen Beyn and  Sergey Piskarev
2008, 10(1): 19-42 doi: 10.3934/dcdsb.2008.10.19 +[Abstract](29) +[PDF](340.3KB)
This paper is devoted to the numerical analysis of abstract semilinear parabolic problems $u'(t) = Au(t) + f(u(t)), u(0)=u^0,$ in some general Banach space $E$. We prove a shadowing Theorem that compares solutions of the continuous problem with those of a semidiscrete approximation (time stays continuous) in the neighborhood of a hyperbolic equilibrium. We allow rather general discretization schemes following the theory of discrete approximations developed by F. Stummel, R.D. Grigorieff and G. Vainikko. We use a compactness principle to show that the decomposition of the flow into growing and decaying solutions persists for this general type of approximation. The main assumptions of our results are naturally satisfied for operators with compact resolvents and can be verified for finite element as well as finite difference methods. In this way we obtain a unified approach to shadowing results derived e.g. in the finite element context ([19, 20, 21]).
On the stopping time of a bouncing ball
Anna Maria Cherubini , Giorgio Metafune and  Francesco Paparella
2008, 10(1): 43-72 doi: 10.3934/dcdsb.2008.10.43 +[Abstract](102) +[PDF](1879.9KB)
We study a simple model of a bouncing ball that takes explicitely into account the elastic deformability of the body and the energy dissipation due to internal friction. We show that this model is not subject to the problem of inelastic collapse, that is, it does not allow an infinite number of impacts in a finite time. We compute asymptotic expressions for the time of flight and for the impact velocity. We also prove that contacts with zero velocity of the lower end of the ball are possible, but non-generic. Finally, we compare our findings with other models and laboratory experiments.
Travelling waves for a combustion model coupled with hyperbolic radiation moment models
Josephus Hulshof and  Pascal Noble
2008, 10(1): 73-90 doi: 10.3934/dcdsb.2008.10.73 +[Abstract](29) +[PDF](228.5KB)
In this paper we consider a model of a flame propagating in a gas containing inert dust. The combustion model is a model with simple chemistry obtained in the high activation energy limit. The radiative field is modeled either by the $P_1$ isotropic model or by the $M_1$ anisotropic model. Under some restrictions on the parameters we prove the existence of travelling waves for this hierarchy of combustion radiation models by Schauder's fixed point argument on bounded domains and uniform estimates.
Homogenization of the Maxwell's system for conducting media
Jiann-Sheng Jiang , Chi-Kun Lin and  Chi-Hua Liu
2008, 10(1): 91-107 doi: 10.3934/dcdsb.2008.10.91 +[Abstract](30) +[PDF](199.1KB)
This paper is devoted to the study of the memory effect induced by homogenization of the Maxwell system for conducting media. The memory kernel is described by the Volterra integral equation. Furthermore, it can be characterized explicitly in terms of Young’s measure, and the kinetic formulation of the homogenized equation is also obtained. The kinetic formulation allows us to obtain the homogenization of the energy density and the associated conservation law with the Poynting vector. The interesting interaction phenomenon of the microscopic and macroscopic scales is also discussed and the memory effect explains qualitatively something about irreversibility.
An investigation of the global properties of a two-dimensional competing species model
Patrick J. Johnson and  Mark E. Burke
2008, 10(1): 109-128 doi: 10.3934/dcdsb.2008.10.109 +[Abstract](32) +[PDF](560.5KB)
In this paper we demonstrate how the global dynamics of a biological model can be analysed. In particular, as an example, we consider a competing species population model based on the discretisation of the original Lotka-Volterra equations. We analyse the local and global dynamic properties of the resulting two-dimensional noninvertible dynamical system in the cases when the interspecific competition is considered to be “weak”, “strong” and “mixed”. The main results of this paper are derived from the study of some global bifurcations that change the structure of the attractors and their basins. These bifurcations are investigated by the use of critical curves, a powerful tool for the analysis of the global properties of noninvertible two-dimensional maps.
Resilience in stem cell renewal: development of the Agur--Daniel--Ginosar model
Oleg U. Kirnasovsky , Yuri Kogan and  Zvia Agur
2008, 10(1): 129-148 doi: 10.3934/dcdsb.2008.10.129 +[Abstract](30) +[PDF](242.7KB)
This work is based on a previous model of Z. Agur, Y. Daniel and Y. Ginosar (2002), retrieving the essential properties of homeostatic tissue development, as reflected by the bone marrow. The original model, represented by cellular automata on a connected, locally finite undirected graph, identifies the minimal basic properties essential for maintaining tissue homeostasis and for guaranteeing the ability of a few stem cells to repopulate the tissue following its depletion. However, this model is too general to ensure a relative “stability” of stem cell numbers in the tissue, a prerequisite for the integrity of biological systems. In the present work, some natural limitations on the model are introduced, under which a formula for the state of a given cell at any given time is obtained, as well as for the proportion of stem cells as a function of model parameters. For tube-like graphs, defined for modeling tissue engineering scaffolds and known tumor geometries, the system obtains a fixed cellular composition, interpreted as homeostasis, thus enabling precise calculation of the necessary conditions for tissue reconstruction. These results also can shed light on conditions for disrupting homeostasis in cancerous tissues.
Stability of travelling waves with noncritical speeds for double degenerate Fisher-Type equations
Yi Li and  Yaping Wu
2008, 10(1): 149-170 doi: 10.3934/dcdsb.2008.10.149 +[Abstract](29) +[PDF](293.3KB)
This paper is concerned with the asymptotic stability of travel- ling wave solutions for double degenerate Fisher-type equations. By spectral analysis, each travelling front solution with non-critical speed is proved to be linearly exponentially stable in some exponentially weighted spaces. Further by Evans function method and detailed semigroup estimates each travelling front solution with non-critical speed is proved to be locally algebraically stable to perturbations in some polynomially weighted spaces, and it is also locally exponentially stable to perturbations in some polynomially and exponentially weighted spaces.
Multi-layer quasi-geostrophic equations of the ocean with delays
T. Tachim Medjo
2008, 10(1): 171-196 doi: 10.3934/dcdsb.2008.10.171 +[Abstract](33) +[PDF](288.7KB)
In this article, we study the multi-layer quasi-geostrophic equations of the ocean with delays. We prove the existence and uniqueness of the solutions to these equations when the external force contains some delays. We also discuss the asymptotic behavior of the solution and the stability of the stationary solutions. Furthermore, we prove in [20] the existence of an attractor for the model.
Boundary stabilization of a nonlinear shallow beam: theory and numerical approximation
Arnaud Münch and  Ademir Fernando Pazoto
2008, 10(1): 197-219 doi: 10.3934/dcdsb.2008.10.197 +[Abstract](32) +[PDF](675.0KB)
We consider a dynamical one-dimensional nonlinear Marguerre-Vlaslov model for an elastic arch depending on a parameter $\varepsilon>0$ and study its asymptotic behavior for large time as $\varepsilon\rightarrow 0$. Introducing appropriate boundary feedbacks, we prove that the corresponding energy decays exponentially as $t\rightarrow \infty$, uniformly with respect to $\varepsilon$ and the curvature. The analysis highlights the importance of the damping mechanism - assumed to be proportional to $\varepsilon^{\alpha}$, $0\leq \alpha\leq 1$ - on the longitudinal deformation of the arch. The limit as $\varepsilon\rightarrow 0$, first exhibits a linear and a nonlinear arch model, for $\alpha>0$ and $\alpha=0$ respectively and then, allows us to obtain exponential decay properties. Some numerical experiments confirm the theoretical results, analyze the cases $\alpha\notin [0,1]$ and evaluate the influence of the curvature on the stabilization.
Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian
Adam M. Oberman
2008, 10(1): 221-238 doi: 10.3934/dcdsb.2008.10.221 +[Abstract](92) +[PDF](2040.6KB)
Certain fully nonlinear elliptic Partial Differential Equations can be written as functions of the eigenvalues of the Hessian. These include: the Monge-Ampère equation, Pucci’s Maximal and Minimal equations, and the equation for the convex envelope. In this article we build convergent monotone finite difference schemes for the aforementioned equations. Numerical results are presented.
A linearly implicit finite difference method for a Klein-Gordon-Schrödinger system modeling electron-ion plasma waves
Pavlos Xanthopoulos and  Georgios E. Zouraris
2008, 10(1): 239-263 doi: 10.3934/dcdsb.2008.10.239 +[Abstract](35) +[PDF](1274.2KB)
An initial and Dirichlet boundary value-problem for a Klein– Gordon–Schrödinger-type system of equations is considered, which describes the nonlinear interaction between high frequency electron waves and low frequency ion plasma waves in a homogeneous magnetic field. To approximate the solution to the problem a linearly implicit finite difference method is proposed, the convergence of which is ensured by deriving a second order error estimate in a discrete energy norm that is stronger than the discrete maximum norm. The numerical implementation of the method gives a computational confirmation of its order of convergence and recovers known theoretical results for the behavior of the solution, while revealing additional nonlinear features.

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