Continuation of the cutoff Boltzmann flow from macroscopic bounds
Let $f \ge 0$ be a smooth solution on $[0, T) \times \mathbb{T}^3_x \times \mathbb{R}^3_v$, $T < \infty$, of the hard-sphere Boltzmann equation with cutoff, $$\partial_t f + v \cdot \nabla_x f = \int_{\mathbb{R}^3} \int_{\mathbb{S}^2} |v - v_*| \, [f(v') f(v_*') - f(v) f(v_*)] \, d\sigma \, dv_*,$$ where $v' = (v + v_*)/2 + |v - v_*|\sigma/2$ and $v_*' = (v + v_*)/2 - |v - v_*|\sigma/2$, and all factors share $(t, x)$. For a range of collision kernels without angular cutoff, Imbert and Silvestre proved that a classical periodic solution whose mass density stays bounded above and below and whose energy and entropy densities stay bounded above is smooth at positive times, with every derivative bounded and decaying faster than any power of $|v|$ (for soft potentials, assuming rapidly decaying initial data). No such regularization is expected with cutoff; Silvestre proposes studying, under the same bounds, the propagation of the smoothness and rapid decay of the initial data.
Problem. Assume that $f(0)$ and all its derivatives decay faster than any inverse power of $|v|$, and that constants $m_0, M_0, E_0 > 0$ and $H_0 \in \mathbb{R}$ satisfy, for all $(t, x)$, $$m_0 \le \int f \, dv \le M_0, \qquad \int |v|^2 f \, dv \le E_0, \qquad \int f \log f \, dv \le H_0.$$ Must $f$ extend beyond $T$ as a smooth solution with the same rapid velocity decay? No a priori bound on $f$ itself or on its derivatives is assumed.
Source: Silvestre (2023), Section 8.5, on propagation of smoothness and rapid decay for the cutoff Boltzmann equation under the hydrodynamic bounds. Stated by AIM — Open Applied Problems as a continuation criterion for hard spheres on the three-dimensional torus.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, Boltzmann equation, conditional regularity
References
@incollection{silvestre2023,
author = {Silvestre, Luis},
title = {Regularity estimates and open problems in kinetic equations},
booktitle = {A{$^3$}{N}{$^2$}{M}: Approximation, Applications, and Analysis of Nonlocal, Nonlinear Models},
series = {The IMA Volumes in Mathematics and its Applications},
publisher = {Springer},
address = {Cham},
year = {2023},
pages = {101--148},
doi = {10.1007/978-3-031-34089-5_3},
eprint = {2204.06401},
archivePrefix = {arXiv},
}
@article{imbert2022,
author = {Imbert, Cyril and Silvestre, Luis},
title = {Global regularity estimates for the {B}oltzmann equation without cut-off},
journal = {J. Amer. Math. Soc.},
volume = {35},
number = {3},
year = {2022},
pages = {625--703},
doi = {10.1090/jams/986},
eprint = {1909.12729},
archivePrefix = {arXiv},
}
@misc{aim-cutoff-boltzmann-continuation,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Continuation of the cutoff Boltzmann flow from macroscopic bounds}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/366-cutoff-boltzmann-continuation.md},
note = {Entry 366 at commit 24edaad, 2026-10-15},
}
Resolution claims
We construct a spatially homogeneous, radial, nonnegative smooth solution of the full-sphere hard-sphere Boltzmann equation whose initial value is a Gaussian but whose velocity tail is a positive multiple of \(r^{-6}\) at every positive time. Mass is one, and energy and entropy have uniform finite upper bounds on a short time interval. This gives a negative answer to a literal …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/main.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00785, "Continuation of the cutoff Boltzmann flow from macroscopic bounds", clopen.solutions.