14 research outputs found
On the convergence to critical scaling profiles in submonolayer deposition models
In this work we study the rate of convergence to similarity profiles in a
mean field model for the deposition of a submonolayer of atoms in a crystal
facet, when there is a critical minimal size for the stability of the
formed clusters. The work complements recently published related results by the
same authors in which the rate of convergence was studied outside of a critical
direction in the cluster size vs. time plane. In this paper
we consider a different similarity variable, ,
corresponding to an inner expansion of that critical direction, and prove the
convergence of solutions to a similarity profile when with fixed, as well as the rate at which the limit is
approached.Comment: Dedicated to the memory of Jack Car
Rates of convergence to scaling profiles in a submonolayer deposition model and the preservation of memory of the initial condition
We establish rates of convergence of solutions to scaling (or similarity)
profiles in a coagulation type system modelling submonolayer deposition. We
prove that, although all memory of the initial condition is lost in the
similarity limit, information about the large cluster tail of the initial
condition is preserved in the rate of approach to the similarity profile. The
proof relies in a change of variables that allows for the decoupling of the
original infinite system of ordinary differential equations into a closed
two-dimensional nonlinear system for the monomer--bulk dynamics and a lower
triangular infinite dimensional linear one for the cluster dynamics. The
detailed knowledge of the long time monomer concentration, which was obtained
earlier by Costin et al. in (O. Costin, M. Grinfeld, K.P. O'Neill and H. Park,
Long-time behaviour of point islands under fixed rate deposition, Commun. Inf.
Syst. 13, (2), (2013), pp.183-200) using asymptotic methods and is rederived
here by center manifold arguments, is then used for the asymptotic evaluation
of an integral representation formula for the concentration of -clusters.
The use of higher order expressions, both for the Stirling expansion and for
the monomer evolution at large times allow us to obtain, not only the
similarity limit, but also the rate at which it is approached.Comment: Revised according to referee's suggestions; to be published in SIAM
J. Math. Ana
Dynamics of a Non-Autonomous ODE System Occurring in Coagulation Theory
We consider a constant coefficient coagulation equation with Becker–D¨oring
type interactions and power law input of monomers J1(t)=αtω, with α >
0 and ω>−1
2 . For this infinite dimensional system we prove solutions converge
to similarity profiles as t and j converge to infinity in a similarity way,
namely with either j/ς or (j −ς)/√ς constants, where ς =ς(t) is a function
of t only. This work generalizes to the non-autonomous case a recent result
of da Costa et al. (2004). Markov Processes Relat. Fields 12, 367–398. and
provides a rigorous derivation of formal results obtained by Wattis J. Phys.
A: Math. Gen. 37, 7823–7841. The main part of the approach is the analysis
of a bidimensional non-autonomous system obtained through an appropriate
change of variables; this is achieved by the use of differential inequalities and
qualitative theory methods. The results about rate of convergence of solutions
of the bidimensional system thus obtained are fed into an integral formula
representation for the solutions of the infinite dimensional system which is
then estimated by an adaptation of methods used by da Costa et al. (2004).
Markov Processes Relat. Fields 12, 367–398.peerreviewe
Convergence to self-similarity in an addition model with power-like time-dependent input of monomers
In this note we extend the results published in Ref. 1 to a coagulation system with Becker-Doring type interactions and time-dependent input of monomers of power–like type: as , with and . The general framework of the proof follows Ref. 1 but a different strategy is needed at a number of points
Scaling behaviour in a coagulation-annihilation model and Lotka-Volterra competition systems
In a recent paper, Laurencot and van Roessel (2010 J. Phys. A: Math. Theor., 43, 455210) studied the scaling behaviour of solutions to a two-species coagulation–annihilation system with total annihilation and equal strength coagulation, and identified cases where self-similar behaviour occurs, and
others where it does not. In this paper, we proceed with the study of this kind
of system by assuming that the coagulation rates of the two different species
need not be equal. By applying Laplace transform techniques, the problem
is transformed into a two-dimensional ordinary differential system that can be
transformed into a Lotka–Volterra competition model. The long-time behaviour
of solutions to this Lotka–Volterra system helps explain the different cases of
existence and nonexistence of similarity behaviour, as well as why, in some
cases, the behaviour is nonuniversal, in the sense of being dependent on initial
conditions.FPC, JTP e RS foram parcialmente financiados pelo CAMGSD-LARSyS através do financiamento plurianual atribuido pela Fundação para a Ciência e Tecnologia (Portugal
Dynamical problems in coagulation equations
Tese de Doutoramento em Matemática apresentada à Universidade AbertaNeste trabalho são analisados alguns aspectos do comportamento asimptótico dos sistemas de um número infinito de equações diferenciais ordinárias que modelam a cinética de partículas
de coagulação dados por
\dot{c}_1 = \alpha t^{\omega} - c_1^2 - c_1 \sum_{j=1}^{\infty} c_j},\dot{c}_j = c_1 c_{j-1} - c_1 c_j, j \geq 2 , onde e são constantes.
Abordamos dois aspectos particularmente importantes do comportamento dinâmico das soluções deste sistema.
Primeiro, o comportamento pontual das soluções quando e o comportamento da quantidade total de agregados definido por .
O segundo aspecto prende-se com a ocorrência de comportamentos auto-semelhantes.
No Capítulo 2 estudamos o caso , no Capítulo 4 o caso
e no no Capítulo 5 o caso utilizando uma mudança de variáveis apropriada.
No Capítulo 3 consideramos uma extensão dos resultados do Capítulo 2, para fontes de monómeros do tipo ,onde é uma função contínua satisfazendo
quando .
Os casos e são tratados no Capítulo 5 utilizando uma abordagem diferente, assente numa análise das propriedades de monotonicidade das soluções.
Os resultados obtidos permitem-nos mostrar a existência de uma função e uma família de funções de escalamento para tais que
se verifica para e .
Resultados semelhantes são também obtidos no caso .
Para o caso alguns resultados parcias, e evidência numérica, sugerem que isso não acontece
Nonlinear Hyperbolic Conservation Laws
The applied problem: Modeling on continuum physics, chemistry, biology, environment, etc.
Areas as gas dynamics, nonlinear elasticity, shallow water theory,
geometric optics, magneto-fluid dynamics, kinetic theory, combustion
theory, cancer medicine, petroleum engineering, irrigation systems, etc.
Applications as optimal shape design (aeronautics, automobiles), noise
reduction in cavities and vehicles, flexible structures, seismic waves
(earthquakes, tsunamis), laser control in quantum mechanical and
molecular systems, chromatography, chemostasis, oil prospection and
recovery, cardiovascular system, traffic flow, the Thames barrier, etc
Modelling silicosis: existence, uniqueness and basic properties of solutions
We present a model for the silicosis disease mechanism following the original proposal by Tran et al. (1995), as modified recently by da Costa et al. (2020). The model consists in an infinite ordinary differential equation system of coagulation–fragmentation–death type. Results of existence, uniqueness, continuous dependence on the initial data and differentiability of solutions are proved for the initial value problem.FCT project CAMGSD UIDB/MAT/04459/2020info:eu-repo/semantics/publishedVersio