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- Provas de Agregação em MatemĂĄtica: lição, relatĂłrio e curriculumPublication . Costa, Fernando Pestana daNuma primeira parte faz-se uma descrição dos tipos de equaçÔes de coagulação-fragmentação mais comuns nas literaturas matemĂĄtica e cientĂfica, referindo-se alguns aspectos histĂłricos considerados relevantes, bem como vĂĄrias ĂĄreas de aplicaçÔes. Na segunda parte descrevem-se resultados matemĂĄticos relativos a existĂȘncia e unicidade de soluçÔes de alguns destes sistemas, nomeadamente os sistemas discretos de Smoluchowski e de coagulação-fragmentação: começando com uma breve apresentação dos espaços funcionais utilizados, passam- se depois em revista os resultados sobre existĂšncia de soluçÔes fornecendo-se uma descrição breve das ideias subjacentes Ă s demonstraçÔes. Esta parte termina com uma secção dedicada aos problemas de unicidade. Nas terceira e quarta partes descrevem-se diversos aspectos do comportamento de soluçÔes. Focam-se com especial atenção questĂ”es sobre a convergĂȘncia para equilĂbrios a tempos longos, sobre o comportamento auto-semelhante de soluçÔes e sobre a conservação, ou nĂŁo conservação, de densidade. Todas estas questĂ”es, alĂ©m da Ăłbvia relevĂąncia matemĂĄtica, tĂȘm tambĂ©m interpretaçÔes fĂsicas de clara importĂąncia para as aplicaçÔes.
- On the positivity of solutions to the Smoluchowski equationsPublication . Costa, Fernando Pestana daThe dynamics of cluster growth can be modelled by the following infinite system of ordinary differential equations, first proposed by Smoluchowski, [8], where cj=cj(t) represents the physical concentration of j-clusters (aggregates of j identical particles), aj,k=aj,kâ„0 are the time-independent coagulation coefficients, measuring the effectiveness of the coagulation process between a j-cluster and a k-cluster, and the first sum in the right-hand side of (1) is defined to be zero if j = 1.
- A finite-dimensional dynamical model for gelation in coagulation processPublication . Costa, Fernando Pestana daWe study a finite-dimensional system of ordinary differential equations derived from Smoluchowskiâs 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 Smoluchowskiâs 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 dynamicsPublication . Costa, Fernando Pestana da; Carr, J.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.
- On the dynamic scaling behaviour of solutions to the discrete Smoluchowski equationsPublication . Costa, Fernando Pestana daIn this paper we generalize recent results of Kreer and Penrose by showing that solutions to the discrete Smoluchowski equations $$\dot{c}_{j} = \sum_{k=1}^{j-1}c_{j-k}c_{k} - 2c_{j}\sum_{k=1}^{\infty}c_{k}, j = 1, 2, \ldots$$ with general exponentially decreasing initial data, with density $\rho,$ have the following asymptotic behaviour $$\lim_{j, t \rightarrow\infty, \xi = j/t fixed, j \in {\cal J}} t^{2}c_{j}(t) = \frac{q}{\rho}\, e^{-\xi/\rho},$$ where ${\cal J} = \{j: c_{j}(t)>0, t>0\}$ and $q =\gcd \{j: c_{j}(0)>0\}.$