Large-data global smoothness for the spatially inhomogeneous Boltzmann equation
On $\mathbb{T}^3_x \times \mathbb{R}^3_v$ consider the Boltzmann equation with the hard-sphere collision kernel with angular cutoff, $$\partial_t f + v \cdot \nabla_x f = Q(f, f), \qquad Q(f, f)(v) = \int_{\mathbb{R}^3} \int_{\mathbb{S}^2} |v - v_*| \bigl[ f(v') f(v_*') - f(v) f(v_*) \bigr] \, d\sigma \, dv_*,$$ where $v' = \frac{v + v_*}{2} + \frac{|v - v_*|}{2} \sigma$ and $v_*' = \frac{v + v_*}{2} - \frac{|v - v_*|}{2} \sigma$, and all factors are evaluated at the same $(t, x)$. DiPerna and Lions constructed global renormalized solutions for cutoff kernels. For kernels without angular cutoff, Henderson, Snelson and Tarfulea constructed classical solutions on a short time interval for large, irregular data, and global classical solutions for data near equilibrium.
Problem. Let $f_0 > 0$ be smooth on $\mathbb{T}^3 \times \mathbb{R}^3$, rapidly decreasing in $v$ together with all its derivatives, and bounded below by $a e^{-b|v|^2}$ for some $a, b > 0$. Does the equation have a unique global smooth solution with $f(0) = f_0$? No closeness of $f_0$ to a Maxwellian is assumed.
Source: A classical question of kinetic theory; see Silvestre (2023), §3. Stated in this form, for the hard-sphere kernel with angular cutoff on the torus, by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, Boltzmann equation, classical solutions, kinetic equations
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},
editor = {Mengesha, Tadele and Salgado, Abner J.},
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{diperna1989,
author = {DiPerna, R. J. and Lions, P.-L.},
title = {On the {C}auchy problem for {B}oltzmann equations: global existence and weak stability},
journal = {Ann. of Math. (2)},
volume = {130},
number = {2},
year = {1989},
pages = {321--366},
doi = {10.2307/1971423},
}
@article{henderson2025,
author = {Henderson, Christopher and Snelson, Stanley and Tarfulea, Andrei},
title = {Classical solutions of the {B}oltzmann equation with irregular initial data},
journal = {Ann. Sci. {\'E}c. Norm. Sup{\'e}r. (4)},
year = {2025},
doi = {10.24033/asens.2603},
eprint = {2207.03497},
archivePrefix = {arXiv},
}
@misc{aim-inhomogeneous-boltzmann-regularity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Large-data global smoothness for the spatially inhomogeneous Boltzmann equation}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/074-inhomogeneous-boltzmann-regularity.md},
note = {Entry 074 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00602, "Large-data global smoothness for the spatially inhomogeneous Boltzmann equation", clopen.solutions.