On the dynamic scaling behaviour of solutions to the discrete Smoluchowski equa...

• Author(s): Costa, Fernando Pestana da
• Date: 1996
• Persistent ID: http://hdl.handle.net/10400.2/1470
• Origin: Repositório Aberto da Universidade Aberta
• Subject(s): Smoluchowski coagulation equations; Self-similar solutions
• Document Type: Article
• Language: English

Description

In 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 $q =\gcd \{j: c_{j}(0)>0\}.$