Convergence to self-similarity for ballistic annihilation dynamics

TL;DR

This paper proves that solutions to the ballistic annihilation Boltzmann equation converge to a universal self-similar profile over time, establishing the rate of convergence and universality of the annihilation process.

Contribution

It demonstrates the convergence to a universal self-similar profile for the ballistic annihilation model and provides explicit algebraic rates of convergence.

Findings

Solutions tend to a self-similar profile over time

Convergence rate is explicitly quantified as algebraic

Universality of the annihilation rate is confirmed

Abstract

We consider the spatially homogeneous Boltzmann equation for ballistic annihilation in dimension d 2. Such model describes a system of ballistic hard spheres that, at the moment of interaction, either annihilate with probability $\alpha$ $\in$ (0, 1) or collide elastically with probability 1 -- $\alpha$. Such equation is highly dissipative in the sense that all observables, hence solutions, vanish as time progresses. Following a contribution , by two of the authors, considering well posedness of the steady self-similar profile in the regime of small annihilation rate $\alpha$ $\ll$ 1, we prove here that such self-similar profile is the intermediate asymptotic attractor to the annihilation dynamics with explicit universal algebraic rate. This settles the issue about universality of the annihilation rate for this model brought in the applied literature.