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Discrete & Continuous Dynamical Systems - B

2006 , Volume 6 , Issue 1

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Homogenization of the Schrödinger equation with a time oscillating potential
Grégoire Allaire and M. Vanninathan
2006, 6(1): 1-16 doi: 10.3934/dcdsb.2006.6.1 +[Abstract](546) +[PDF](283.6KB)
Abstract:
We study the homogenization of a Schrödinger equation in a periodic medium with a time dependent potential. This is a model for semiconductors excited by an external electromagnetic wave. We prove that, for a suitable choice of oscillating (both in time and space) potential, one can partially transfer electrons from one Bloch band to another. This justifies the famous "Fermi golden rule" for the transition probability between two such states which is at the basis of various optical properties of semiconductors. Our method is based on a combination of classical homogenization techniques (two-scale convergence and suitable oscillating test functions) and of Bloch waves theory.
Meshfree method for the non-stationary incompressible Navier-Stokes equations
Hi Jun Choe, Do Wan Kim and Yongsik Kim
2006, 6(1): 17-39 doi: 10.3934/dcdsb.2006.6.17 +[Abstract](509) +[PDF](729.5KB)
Abstract:
We consider the solvability and the error estimates of numerical solutions of the non-stationary incompressible Stokes and Navier-Stokes equations by the meshfree method. The moving least square reproducing kernel method or the MLSRK method is employed for the space approximations. The existence of numerical solutions and the $L^2$-type error estimates are obtained. As a numerical example, we compare the numerical solutions of the Stokes and the Navier-Stokes equations with the exact solutions. Also we solve the non-stationary Navier-Stokes driven cavity flow using the MLSRK method.
Stabilized finite element method for the non-stationary Navier-Stokes problem
Yinnian He, Yanping Lin and Weiwei Sun
2006, 6(1): 41-68 doi: 10.3934/dcdsb.2006.6.41 +[Abstract](702) +[PDF](350.9KB)
Abstract:
In this article, a locally stabilized finite element formulation of the two-dimensional Navier-Stokes problem is used. A macroelement condition which provides the stability of the $Q_1-P_0$ quadrilateral element and the $P_1-P_0$ triangular element is introduced. Moreover, the $H^1$ and $L^2$-error estimates of optimal order for finite element solution $(u_h,p_h)$ are analyzed. Finally, a uniform $H^1$ and $L^2$-error estimates of optimal order for finite element solution $(u_h,p_h)$ is obtained if the uniqueness condition is satisfied.
Mathematical analysis of an age-structured SIR epidemic model with vertical transmission
Hisashi Inaba
2006, 6(1): 69-96 doi: 10.3934/dcdsb.2006.6.69 +[Abstract](459) +[PDF](330.2KB)
Abstract:
In this paper, we consider a mathematical model for the spread of a directly transmitted infectious disease in an age-structured population. We assume that infected population is recovered with permanent immunity or quarantined by an age-specific schedule, and the infective agent can be transmitted not only horizontally but also vertically from adult individuals to their newborns. For simplicity we assume that the demographic process of the host population is not affected by the spread of the disease, hence the host population is a demographic stable population. First we establish the mathematical well-posedness of the time evolution problem by using the semigroup approach. Next we prove that the basic reproduction ratio is given as the spectral radius of a positive operator, and an endemic steady state exists if and only if the basic reproduction ratio $R_0$ is greater than unity, while the disease-free steady state is globally asymptotically stable if $R_0 < 1$. We also show that the endemic steady states are forwardly bifurcated from the disease-free steady state when $R_0$ crosses the unity. Finally we examine the conditions for the local stability of the endemic steady states.
A global attractivity result for maps with invariant boxes
M. R. S. Kulenović and Orlando Merino
2006, 6(1): 97-110 doi: 10.3934/dcdsb.2006.6.97 +[Abstract](531) +[PDF](270.5KB)
Abstract:
We present a global attractivity result for maps generated by systems of autonomous difference equations. It is assumed that the map of the system leaves invariant a box, is monotone in a coordinate-wise sense (but not necessarily monotone with respect to a standard cone), and satisfies certain algebraic condition. It is shown that there exists a unique equilibrium, and that it is a global attractor. As an application, it is shown that a discretized version of the Lotka-Volterra system of differential equations of order $k$ has a global attractor in the positive orthant for certain range of parameters.
On a well-posed turbulence model
W. Layton and R. Lewandowski
2006, 6(1): 111-128 doi: 10.3934/dcdsb.2006.6.111 +[Abstract](472) +[PDF](247.9KB)
Abstract:
This report considers mathematical properties, important for practical computations, of a model for the simulation of the motion of large eddies in a turbulent flow. In this model, closure is accomplished in the very simple way:

$\overline{u u} $˜ $\overline{\bar {u} \bar {u}}$, yielding the model
$\nabla \cdot w= 0, \quad w_{t} + \nabla \cdot (\overline{w w}) - \nu \Delta w + \nabla q = \bar {f}$.

In particular, we prove existence and uniqueness of strong solutions, develop the regularity of solutions of the model and give a rigorous bound on the modelling error, $||\bar {u} - w||$. Finally, we consider the question of non-physical vortices (false eddies), proving that the model correctly predicts that only a small amount of vorticity results when the total turning forces on the flow are small.

Drug resistance in cancer chemotherapy as an optimal control problem
Urszula Ledzewicz and Heinz Schättler
2006, 6(1): 129-150 doi: 10.3934/dcdsb.2006.6.129 +[Abstract](667) +[PDF](436.7KB)
Abstract:
We analyze non cell-cycle specific mathematical models for drug resistance in cancer chemotherapy. In each model developing drug resistance is inevitable and the issue is how to prolong its onset. Distinguishing between sensitive and resistant cells we consider a model which includes interactions of two killing agents which generate separate resistant populations. We formulate an associated optimal control problem for chemotherapy and analyze the qualitative structure of corresponding optimal controls.
Traveling waves in compressible elastic rods
Jonatan Lenells
2006, 6(1): 151-167 doi: 10.3934/dcdsb.2006.6.151 +[Abstract](391) +[PDF](361.4KB)
Abstract:
We determine all weak traveling wave solutions of a model for nonlinear dispersive waves in cylindrical compressible hyperelastic rods. Besides the previously known smooth, peaked, and cusped solutions, the equation is shown to admit compactons, stumpons, and fractal-like waves.
Block structure and block stability of two-dimensional incompressible flows
Tian Ma and Shouhong Wang
2006, 6(1): 169-184 doi: 10.3934/dcdsb.2006.6.169 +[Abstract](457) +[PDF](263.7KB)
Abstract:
We study in this article topological structure of divergence-free vector fields on general two-dimensional manifolds. We introduce a new concept called block structural stability (or block stability for simplicity) and prove that the block stable divergence-free vector fields form a dense and open set. Furthermore, we show that a block stable divergence-free vector field, which we call a basic vector field, is fully characterized by a nice and simple structure, which we call block structure. The results and ideas presented in this article have been applied to studies on structure and its evolutions of the solutions of the Navier-Stokes equations; see [4, 9, 10].
A model for disease transmission in a patchy environment
Mahin Salmani and P. van den Driessche
2006, 6(1): 185-202 doi: 10.3934/dcdsb.2006.6.185 +[Abstract](549) +[PDF](343.7KB)
Abstract:
For a spatially heterogeneous environment with patches in which travel rates between patches depend on disease status, a disease transmission model is formulated as a system of ordinary differential equations. An expression for the basic reproduction number $R_0$ is derived, and the disease free equilibrium is shown to be globally asymptotically stable for $R_0<1$. Easily computable bounds on $R_0$ are derived. For a disease with very short exposed and immune periods in an environment with two patches, the model is analyzed in more detail. In particular, it is proved that if susceptible and infectious individuals travel at the same rate, then $R_0$ gives a sharp threshold with the endemic equilibrium being globally asymptotically stable for $R_0>1$. If parameters are such that for isolated patches the disease is endemic in one patch but dies out in the other, then travel of infectious individuals from the patch with endemic disease may lead to the disease becoming endemic in both patches. However, if this rate of travel is increased, then the disease may die out in both patches. Thus travel of infectious individuals in a patchy environment can have an important influence on the spread of disease.
Cauchy problem for the nonlinear Schrödinger-IMBq equations
Shubin Wang and Guowang Chen
2006, 6(1): 203-214 doi: 10.3934/dcdsb.2006.6.203 +[Abstract](454) +[PDF](229.0KB)
Abstract:
In this paper, the existence and uniqueness of the global strong solution of the Cauchy problem for nonlinear Schrödinger-IMBq equations are proved.
Best response dynamics for continuous zero--sum games
Josef Hofbauer and Sylvain Sorin
2006, 6(1): 215-224 doi: 10.3934/dcdsb.2006.6.215 +[Abstract](359) +[PDF](223.0KB)
Abstract:
We study best response dynamics in continuous time for continuous concave-convex zero-sum games and prove convergence of its trajectories to the set of saddle points, thus providing a dynamical proof of the minmax theorem. Consequences for the corresponding discrete time process with small or diminishing step-sizes are established, including convergence of the fictitious play procedure.
Analysis of a model for the dynamics of prions
Jan Prüss, Laurent Pujo-Menjouet, G.F. Webb and Rico Zacher
2006, 6(1): 225-235 doi: 10.3934/dcdsb.2006.6.225 +[Abstract](630) +[PDF](462.9KB)
Abstract:
A mathematical model for the dynamics of prion proliferation is analyzed. The model involves a system of three ordinary differential equations for the normal prion forms, the abnormal prion forms, and polymers comprised of the abnormal forms. The model is a special case of a more general model, which is also applicable to other models of infectious diseases. A theorem of threshold type is derived for this general model. It is proved that below and at the threshold, there is a unique steady state, the disease-free equilibrium, which is globally asymptotically stable. Above the threshold, the disease-free equilibrium is unstable, and there is another steady state, the disease equilibrium, which is globally asymptotically stable.

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