Abstract:We prove that linear collisional kinetic equations in the whole space without confinement mechanism display a long-time self-similar behaviour.
This drastically improves the recently known results (decay estimates) about the solutions in such a context, providing the first result regarding this self-similar behaviour. As a consequence, we also establish a uniform-in-time convergence of the suitably rescaled solutions to their diffusion limit, which is also new.
The class of equations considered includes some BGK type equations, some kinetic nonlocal Fokker--Planck-type equations and
some kinetic (possibly fractional) Fokker--Planck equations, for which we are able to write explicitly solutions through a Wild sum (or Dyson series)
or we can manage some accurate computations on the Fourier side.
| Comments: | 55 pages |
| Subjects: | Analysis of PDEs (math.AP); Probability (math.PR) |
| MSC classes: | 35B40 (Primary) 82C40, 35R11, 45M05, 60E15, 60K35 (Secondary) |
| Cite as: | arXiv:2605.24685 [math.AP] |
| (or arXiv:2605.24685v1 [math.AP] for this version) | |
| https://doi.org/10.48550/arXiv.2605.24685 arXiv-issued DOI via DataCite (pending registration) |
From: Niccolò Tassi [view email]
[v1]
Sat, 23 May 2026 17:45:37 UTC (311 KB)
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