Landau–Coulomb smoothing controlled only by macroscopic quantities
Consider nonnegative smooth rapidly decaying solutions on $[0, T] \times \mathbb{R}^3$ of the spatially homogeneous Landau equation with Coulomb potential, $$\partial_t f = \nabla_v \cdot \bigl(A[f] \nabla_v f - f \nabla_v a[f]\bigr), \qquad A[f](v) = \frac{1}{8\pi} \int_{\mathbb{R}^3} \frac{I - \widehat{v - w} \otimes \widehat{v - w}}{|v - w|} f(w) \, dw, \qquad a[f](v) = \frac{1}{4\pi} \int_{\mathbb{R}^3} \frac{f(w)}{|v - w|} \, dw,$$ where $\widehat{z} = z / |z|$, normalized by $\int f \, dv = 1$, $\int v f \, dv = 0$ and $\int |v|^2 f \, dv = 3$. Golding, Gualdani and Loher constructed global smooth solutions for initial data in a weighted $L^{3/2}$ space with a polynomial lower bound, satisfying $\|f(t)\|_{L^\infty} \le C/t$ for $0 < t \le T_0$, with $T_0 > 0$ and $C$ depending on the initial datum. Ji proved that the Fisher information satisfies $\int |\nabla_v f|^2 / f \, dv \le C_0 (1 + 1/t)$ for $0 < t \le T$, with $C_0$ depending on $T$, the mass, the entropy and $\int f(0,v) |v|^5 \, dv$, and deduced a bound on $\sup_{\tau \le t \le T} \|f(t)\|_{L^\infty}$ with the same dependence and on $\tau$.
Problem. For every $H \in \mathbb{R}$ and $0 < \tau < T$, is there a finite $C(H, \tau, T)$ such that every such solution with $\int f(0,v) \log f(0,v) \, dv \le H$ satisfies $$\sup_{\tau \le t \le T} \|f(t)\|_{L^\infty_v} \le C(H, \tau, T)?$$ The constant must be independent of all initial higher moments, Fisher information, pointwise bounds and derivative norms. No sharp rate in $\tau$ is asked for.
Source: Golding, Gualdani and Loher (2025), the first of their open problems, which also asks whether the conjectured rate 1/t holds; stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, Landau equation, regularization estimates
References
@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},
}
@misc{ji2024,
author = {Ji, Sehyun},
title = {Dissipation estimates of the {F}isher information for the {L}andau equation},
year = {2024},
eprint = {2410.09035},
archivePrefix = {arXiv},
}
@misc{aim-landau-macroscopic-smoothing,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Landau–Coulomb smoothing controlled only by macroscopic quantities}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/364-landau-macroscopic-smoothing.md},
note = {Entry 364 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-00608, "Landau–Coulomb smoothing controlled only by macroscopic quantities", clopen.solutions.