F. P. da Costa

A Finite-Dimensional Dynamical Model for Gelation in Coagulation Processes

Summary. We study a finite-dimensional system of ordinary differential equations derived from Smoluchowskis coagulation equations and whose solutions mimic the behaviour of the nondensity-conserving (geling) solutions in those equations. The analytic and numerical studies of the finite-dimensional system reveals an interesting dynamic behaviour in several respects: Firstly, it suggests that some special geling solutions to Smoluchowskis equations discovered by Leyvraz can have an important dynamic role in gelation studies, and, secondly, the dynamics is interesting in its own right with an attractor possessing an unexpected structure of equilibria and connecting orbits.

Instantaneous gelation in coagulation dynamics

The coagulation equations are a model for the dynamics of cluster growth in which clusters can coagulate via binary interactions to form larger clusters. For a certain class of rate coefficients we prove that the density is not conserved on any time interval.

Asymptotic behavior of solutions to the Coagulation-Fragmentation Equations. II. Weak Fragmentation

The discrete coagulation-fragmentation equations are a model for the kinetics of cluster growth in which clusters can coagulate via binary interactions to form larger clusters or fragment to form smaller ones. The assumptions made on the fragmentation coefficients have the physical interpretation that surface effects are important. Our results on the asymptotic behavior of solutions generalize the corresponding results of Ball, Carr, and Penrose for the Becker-Doring equation.

On the Dynamic Scaling Behaviour of Solutions to the Discrete Smoluchowski Equations

In this paper we generalize recent results of Kreer and Penrose by showing that solutions to the discrete Smoluchowski equations ċj = j−1 ∑ k=1 cj−kck − 2cj ∞ ∑ k=1 ck, j = 1, 2, . . . with general exponentially decreasing initial data, with density ρ, have the following asymptotic behaviour lim j, t→∞ ξ=j/t fixed j∈J tcj(t) = q ρ e−ξ/ρ, where J = {j : cj(t) > 0, t > 0} and q = gcd{j : cj(0) > 0}. AMS Subject Classification (1991): 82C22, 34D05, 12D10

Unimodality of steady size distributions of growing cell populations

Abstract. We consider an equation for the evolution of growing and dividing cells, and show, using a result of Kato and McLeod, that the probability density function for the stationary size distribution is necessarily unimodal.

Scaling behaviour in a coagulation?annihilation model and Lotka?Volterra competition systems

FPC, JTP e RS foram parcialmente financiados pelo CAMGSD-LARSyS atraves do financiamento plurianual atribuido pela Fundacao para a Ciencia e Tecnologia (Portugal)

Mathematical Aspects of Coagulation-Fragmentation Equations

We give an overview of the mathematical literature on the coagulation-like equations, from an analytic deterministic perspective. In Sect. 1 we present the coagulation type equations more commonly encountered in the scientific and mathematical literature and provide a brief historical overview of relevant works. In Sect. 2 we present results about existence and uniqueness of solutions in some of those systems, namely the discrete Smoluchowski and coagulation-fragmentation: we start by a brief description of the function spaces, and then review the results on existence of solutions with a brief description of the main ideas of the proofs. This part closes with the consideration of uniqueness results. In Sects. 3 and 4 we are concerned with several aspects of the solutions behaviour. We pay special attention to the long time convergence to equilibria, self-similar behaviour, and density conservation or lack thereof.

Kickback in nematic liquid crystals

We describe a nonlocal linear partial differential equation arising in the analysis of dynamics of a nematic liquid crystal. We confirm that it accounts for the kickback phenomenon by decoupling the director dynamics from the flow. We also analyse some of the mathematical properties of the decoupled director equation.

Self-similar Behaviour in an Addition Model with Input of Monomers

We consider a constant coefficient coagulation equation with Becker‐Doring type interactions and power law input of monomers J1(t) = αtω, with ω∈R. For ω⩾−1/2 we prove solutions converge to similarity profiles.

A MATHEMATICAL STUDY OF A BISTABLE NEMATIC LIQUID CRYSTAL DEVICE

We consider a model of a bistable nematic liquid crystal device based on the Ericksen–Leslie theory. The resulting mathematical object is a parabolic PDE with nonlinear dynamic boundary conditions. We analyze well-posedness of the problem and global existence of solutions using the theory developed by Amann. Furthermore, using phase-plane methods, we give an exhaustive description of the steady state solutions and hence of the switching capabilities of the device.