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Volume 1, 2001

Discrete & Continuous Dynamical Systems - B

2001 , Volume 1 , Issue 4

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Partially saturated flow in a poroelastic medium
R.E. Showalter and Ning Su
2001, 1(4): 403-420 doi: 10.3934/dcdsb.2001.1.403 +[Abstract](96) +[PDF](234.9KB)
Abstract:
The formulation and existence theory is presented for a system modeling diffusion of a slightly compressible fluid through a partially saturated poroelastic medium. Nonlinear effects of density, saturation, porosity and permeability variations with pressure are included, and the seepage surface is determined by a variational inequality on the boundary.
Synchronization in directionally coupled systems: Some rigorous results
V. Afraimovich, J.-R. Chazottes and A. Cordonet
2001, 1(4): 421-442 doi: 10.3934/dcdsb.2001.1.421 +[Abstract](166) +[PDF](254.9KB)
Abstract:
We prove, under some general assumptions, that master-slave synchronization implies generalized synchronization, that is we show the existence and continuity of the functional dependence between the “slave” coordinates and the “master” ones. Then, we prove that this function may be Lipschitz continuous and even less “smooth”, that is only Hölder continuous, depending on the coupling strength. We go beyond the above mentioned assumptions by coupling two identical maps of the interval that are neither continuous nor invertible to prove ‘almost-everywhere’ synchronization instead of global synchronization. Then we relate the Hausdorff dimension and the dimension for Poincaré recurrence of the attractor of master and slave systems. We provide some examples illustrating these results.
An infinite horizon optimal control problem for some switching systems
Fabio Bagagiolo
2001, 1(4): 443-462 doi: 10.3934/dcdsb.2001.1.443 +[Abstract](79) +[PDF](223.6KB)
Abstract:
We study an infinite horizon optimal control problem for a system with two state variables. They are respectively the input and the output of either a delayed relay or a finite sum of delayed relays. The system exhibits hysteresis and the output of the relay is discontinuous. We consider two different switching rules for the relay. We characterize the value functions by the unique viscosity solution of some suitably coupled Dirichlet problems for Hamilton-Jacobi equations.
Structural stability and continuous dependence of solutions of thermoelasticity of type III
Ramon Quintanilla
2001, 1(4): 463-470 doi: 10.3934/dcdsb.2001.1.463 +[Abstract](86) +[PDF](122.2KB)
Abstract:
We prove a structural stability result on the coupling coefficients and continuous dependence on the external data in the thermoelastic theory called of type III. The only condition we require on the elasticity tensor is the symmetry of the coefficients. We use logarithmic convexity arguments.
A size structured model of cell dwarfism
Keith E. Howard
2001, 1(4): 471-484 doi: 10.3934/dcdsb.2001.1.471 +[Abstract](94) +[PDF](163.1KB)
Abstract:
This work is devoted to analyzing the growth of a cell population where cellular development is characterized by cellular size. By size we mean some quantifiedmeasure of cellular mass. The issues treatedhere include: (1) existence anduniqueness of solutions, (2) stability andinstability of solutions, and (3) biological interpretation of results.
On Maxwell's system with a thermal effect
J. J. Morgan and Hong-Ming Yin
2001, 1(4): 485-494 doi: 10.3934/dcdsb.2001.1.485 +[Abstract](90) +[PDF](183.9KB)
Abstract:
In this paper we study Maxwell’s system coupled with a heat equation in one space dimension. The system models a microwave heating process. The feature of the model is that the electric conductivity $\sigma(u)$ strongly depends on the temperature. It is shown that the system has a global solution for $\sigma(u)=1+u^k$ with any $k\ge 1$. The long time behavior of the solution is also investigated.
Meshless method for the stationary incompressible Navier-Stokes equations
Hi Jun Choe, Hyea Hyun Kim, Do Wan Kim and Yongsik Kim
2001, 1(4): 495-526 doi: 10.3934/dcdsb.2001.1.495 +[Abstract](174) +[PDF](500.4KB)
Abstract:
Mathematical analysis is achieved on a meshless method for the stationary incompressible Stokes and Navier-Stokes equations. In particular, the Moving Least Square Reproducing Kernel (MLSRK) method is employed. The existence of discrete solution and its error estimate are obtained. As a numerical example for convergence analysis, we compute the numerical solutions for these equations to compare with exact solutions. Also we solve the driven cavity flow numerically as a test problem.
The stochastic Landau equation as an amplitude equation
D. Blömker, S. Maier-Paape and G. Schneider
2001, 1(4): 527-541 doi: 10.3934/dcdsb.2001.1.527 +[Abstract](138) +[PDF](189.6KB)
Abstract:
We consider a stochastic partial differential equation (Swift-Hohenberg equation) on the real axis with periodic boundary conditions that arises in pattern formation. If the trivial solution is near criticality, and if the stochastic forcing and the deterministic (in)stability are of a comparable magnitude, a so called stochastic Landau equation can be derived in order to describe the dynamics of the bifurcating solutions. Here we establish attractivity and approximation results for this equation.

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