Convergence to self-similarity for ballistic annihilation dynamics. (English. French summary) Zbl 1442.82029
Summary: We consider the spatially homogeneous Boltzmann equation for ballistic annihilation in dimension \(d\geqslant 2\). 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.
MSC:
| 82C40 | Kinetic theory of gases in time-dependent statistical mechanics |
| 74H40 | Long-time behavior of solutions for dynamical problems in solid mechanics |
| 35C06 | Self-similar solutions to PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35Q20 | Boltzmann equations |
Keywords:
ballistic annihilation; reacting particles; self-similarity; long-time asymptotic; annihilation rateReferences:
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