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 $j$-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

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 $n\geq 2$ 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 $x=\tau$ in the cluster size $x$ vs. time $\tau$ plane. In this paper we consider a different similarity variable, $\xi := (x-\tau)/\sqrt{\tau}$, corresponding to an inner expansion of that critical direction, and prove the convergence of solutions to a similarity profile $\Phi_{2,n}(\xi)$ when $x, \tau\to +\infty$ with $\xi$ fixed, as well as the rate at which the limit is approached.Comment: Dedicated to the memory of Jack Car

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 $J_{1}(t)$ of power–like type: $J_{1}(t)/(\alpha t^{\omega }) \rightarrow 1$ as $t \rightarrow \infty$, with $\alpha > 0$ and $\omega > − \frac{1}{2}$. 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 $\alpha>0$ e $\omega$ são constantes. Abordamos dois aspectos particularmente importantes do comportamento dinâmico das soluções deste sistema. Primeiro, o comportamento pontual das soluções quando $t \rightarrow +\infty$ e o comportamento da quantidade total de agregados definido por $\sum_{j=1}^{\infty} c_j$. O segundo aspecto prende-se com a ocorrência de comportamentos auto-semelhantes. No Capítulo 2 estudamos o caso $\omega > -1/2$ , no Capítulo 4 o caso $\omega = -1/2$ e no no Capítulo 5 o caso $\omega < -1/2$ utilizando uma mudança de variáveis apropriada. No Capítulo 3 consideramos uma extensão dos resultados do Capítulo 2, para fontes de monómeros do tipo $J_1 (t)=\alpha t^\omega (1+\varepsilon (t))$,onde $\varepsilon (\cdot)$ é uma função contínua satisfazendo $\varepsilon (t) \to 0$ quando $t \to +\infty$. Os casos $-1 < \omega < -1/2$ e $\omega < -1$ 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 $\varsigma (t) \sim t^{\frac{\omega+2}{3}}$ e uma família de funções de escalamento $\Phi_{1,\omega}$ para $\omega > -\frac{1}{2}$ tais que $c_j(t) \sim \varsigma (t)^{-a} \Phi(j \varsigma (t)^{-b})$ se verifica para $a=\frac{1-\omega}{2+\omega}$ e $b=1$. Resultados semelhantes são também obtidos no caso $\omega = -\frac{1}{2}$. Para o caso $\omega < -\frac{1}{2}$ 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