Existence of self-similar profile for a kinetic annihilation model

TL;DR

This paper proves the existence of a self-similar solution for a modified Boltzmann equation modeling probabilistic ballistic annihilation, where particles either annihilate or collide elastically, with non-conservation of particles, momentum, and energy.

Contribution

It demonstrates the existence of self-similar profiles in a kinetic model with probabilistic annihilation, expanding understanding of non-conservative particle systems.

Findings

Self-similar solutions exist for annihilation probability below a threshold.

The model describes a system where particles can annihilate or collide elastically.

Existence is established for a range of the annihilation parameter .

Abstract

We show the existence of a self-similar solution for a modified Boltzmann equation describing probabilistic ballistic annihilation. Such a model describes a system of hard-spheres such that, whenever two particles meet, they either annihilate with probability $\alpha \in (0,1)$ or they undergo an elastic collision with probability $1 - \alpha$. For such a model, the number of particles, the linear momentum and the kinetic energy are not conserved. We show that, for $\alpha$ smaller than some explicit threshold value $ \alpha_*$, a self-similar solution exists.