Are bounded ancient Landau–Coulomb distributions necessarily Maxwellians?
The spatially homogeneous Landau equation with Coulomb potential on $\mathbb{R}^3$ is $$\partial_t f = \nabla_v \cdot \bigl(A[f] \nabla_v f - f \nabla_v a[f]\bigr), \qquad A[f](v) = \frac{1}{8\pi} \int_{\mathbb{R}^3} \frac{I - \widehat{v - w} \otimes \widehat{v - w}}{|v - w|} f(w) \, dw, \qquad a[f](v) = \frac{1}{4\pi} \int_{\mathbb{R}^3} \frac{f(w)}{|v - w|} \, dw,$$ where $\widehat{z} = z / |z|$. Guillen and Silvestre proved that the Fisher information is nonincreasing along its solutions, and obtained global classical solutions for $C^1$ initial data with Maxwellian upper bounds. Silvestre arrives at the problem below through a blow-up analysis, whose limit profiles are bounded but need not have finite mass or energy; the integrability condition makes the coefficients $A[f]$ and $a[f]$ finite.
Problem. Let $f : (-\infty, 0] \times \mathbb{R}^3 \to [0, 1]$ be a smooth classical solution such that $\int_{\mathbb{R}^3} f(t,v) (1 + |v|^2)^{-1/2} \, dv < \infty$ for every $t \le 0$. Must $f$ be stationary and equal to $A_0 e^{-b|v - c|^2}$ for some constants $A_0 \ge 0$, $b > 0$ and $c \in \mathbb{R}^3$? Finite mass, energy and entropy are not assumed; the zero solution is included.
Source: Silvestre (2023), Conjecture 7.5; see also Golding, Gualdani and Loher (2025), the second of their open problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, ancient solutions, Landau equation, Liouville theorems
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},
volume = {165},
publisher = {Springer},
address = {Cham},
year = {2023},
pages = {101--148},
doi = {10.1007/978-3-031-34089-5_3},
eprint = {2204.06401},
archivePrefix = {arXiv},
}
@article{golding2025,
author = {Golding, William and Gualdani, Maria and Loher, Am{\'e}lie},
title = {Global smooth solutions to the {L}andau--{C}oulomb equation in {$L^{3/2}$}},
journal = {Arch. Ration. Mech. Anal.},
volume = {249},
number = {3},
year = {2025},
pages = {Paper No. 34},
doi = {10.1007/s00205-025-02107-x},
eprint = {2401.06939},
archivePrefix = {arXiv},
}
@article{guillen2025,
author = {Guillen, Nestor and Silvestre, Luis},
title = {The {L}andau equation does not blow up},
journal = {Acta Math.},
volume = {234},
number = {2},
year = {2025},
pages = {315--375},
doi = {10.4310/ACTA.2025.v234.n2.a2},
eprint = {2311.09420},
archivePrefix = {arXiv},
}
@misc{aim-landau-ancient-liouville,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Are bounded ancient Landau–Coulomb distributions necessarily Maxwellians?}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/363-landau-ancient-liouville.md},
note = {Entry 363 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-00607, "Are bounded ancient Landau–Coulomb distributions necessarily Maxwellians?", clopen.solutions.