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 v^{−σ}. The validity of the equation has been rigorously proved in  taking as a starting point a particle model and for values of the exponent σ &gt; 3, but the model can be expected to be valid, on heuristic grounds, for σ &gt; 5/3. 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 5/3&lt; σ &lt; 2 the linear Smoluchowski equation is well-posed and that there exists a unique self-similar profile which is asymptotically stable.

### Self-similar asymptotic behavior for the solutions of a linear coagulation equation

#### 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 v^{−σ}. The validity of the equation has been rigorously proved in  taking as a starting point a particle model and for values of the exponent σ > 3, but the model can be expected to be valid, on heuristic grounds, for σ > 5/3. 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 5/3< σ < 2 the linear Smoluchowski equation is well-posed and that there exists a unique self-similar profile which is asymptotically stable.
##### Scheda breve Scheda completa Scheda completa (DC)
2019
File in questo prodotto:
Non ci sono file associati a questo prodotto.
##### Pubblicazioni consigliate

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11697/151267`
##### Citazioni
• ND
• 7
• 6