Self-similar asymptotic behavior for the solutions of a linear coagulation equation |
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Authors: | Barbara Niethammer Alessia Nota Sebastian Throm Juan J.L. Velázquez |
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Affiliation: | 1. University of Bonn, Institute for Applied Mathematics, Endenicher Allee 60, D-53115 Bonn, Germany;2. Technical University of Munich, Faculty of Mathematics, Boltzmannstrasse 3, D-85748 Garching bei München, Germany |
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Abstract: | In this paper we consider the long-time asymptotics of a linear version of the Smoluchowski equation which describes the evolution of a tagged particle moving in a random distribution of fixed particles. The volumes v of these particles are independently distributed according to a probability distribution which decays asymptotically as a power law . The validity of the equation has been rigorously proved in [22] taking as a starting point a particle model and for values of the exponent , but the model can be expected to be valid, on heuristic grounds, for . The resulting equation is a non-local linear degenerate parabolic equation. The solutions of this equation display a rich structure of different asymptotic behaviors according to the different values of the exponent σ. Here we show that for the linear Smoluchowski equation is well-posed and that there exists a unique self-similar profile which is asymptotically stable. |
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Keywords: | Linear Smoluchowski's equation Coagulation dynamics Long-time asymptotics Self-similar profiles |
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