On a Model for Mass Aggregation with Maximal Size

TL;DR

This paper investigates a kinetic model for particle coagulation with size constraints, analyzing self-similar solutions and long-term behavior through rigorous proofs and numerical simulations, revealing different regimes based on a key parameter.

Contribution

It introduces a new kinetic mean-field model with size-dependent coagulation rules and provides rigorous analysis of existence, stability, and asymptotic behavior of solutions.

Findings

For const>2, two self-similar solutions exist, with one being stable.

Numerical simulations show solutions tend to the stable self-similar solution for const>2.

For const<2, solutions do not exhibit self-similarity and depend on initial data.

Abstract

We study a kinetic mean-field equation for a system of particles with different sizes, in which particles are allowed to coagulate only if their sizes sum up to a prescribed time-dependent value. We prove well-posedness of this model, study the existence of self-similar solutions, and analyze the large-time behavior mostly by numerical simulations. Depending on the parameter $\Dconst$, which controls the probability of coagulation, we observe two different scenarios: For $\Dconst>2$ there exist two self-similar solutions to the mean field equation, of which one is unstable. In numerical simulations we observe that for all initial data the rescaled solutions converge to the stable self-similar solution. For $\Dconst<2$, however, no self-similar behavior occurs as the solutions converge in the original variables to a limit that depends strongly on the initial data. We prove rigorously a corresponding statement for $\Dconst\in (0,1/3)$. Simulations for the cross-over case $\Dconst=2$ are not completely conclusive, but indicate that, depending on the initial data, part of the mass evolves in a self-similar fashion whereas another part of the mass remains in the small particles.