Global classical solutions of the spatially inhomogeneous Landau–Coulomb equation
Let $f(t, x, v) \ge 0$, with $x \in \mathbb{T}^3 = (\mathbb{R}/\mathbb{Z})^3$ and $v \in \mathbb{R}^3$, solve the Landau equation with Coulomb potential and no external or self-consistent force, $$\partial_t f + v \cdot \nabla_x f = \nabla_v \cdot \int_{\mathbb{R}^3} a(v - w) \bigl[ f(t,x,w) \nabla_v f(t,x,v) - f(t,x,v) \nabla_w f(t,x,w) \bigr] \, dw, \qquad a(z) = \frac{1}{|z|} \Bigl( I - \frac{z \otimes z}{|z|^2} \Bigr).$$ Let $f_0 \ge 0$ be smooth with $\sup_{x,v} \langle v \rangle^m |\partial_x^\alpha \partial_v^\beta f_0(x,v)| < \infty$ for every $m \ge 0$ and all multi-indices $\alpha, \beta$, where $\langle v \rangle = (1 + |v|^2)^{1/2}$, and with $f_0(x, v) \ge \delta$ for every $x$ and every $|v| \le r$, for some $\delta, r > 0$. No smallness or closeness to a Maxwellian is assumed. Guo constructed global classical solutions in the periodic box for data close to a Maxwellian. For large data of this kind, Henderson, Snelson and Tarfulea constructed a local classical solution, unique among uniformly continuous weak solutions with enough velocity decay, which can be continued as long as its mass density and its $L^\infty$ norm stay bounded. Snelson and Solomon showed that boundedness, uniformly in $t$ and $x$, of the mass density, of a velocity moment of some positive order and of the $L^p_v$ norm for some $p > 3/2$ suffices, and Golding, Henderson and Silvestre that, on $\mathbb{T}^3$, the solution continues as long as $\int_0^T \|f(t)\|_{L^\infty} \, dt$ is finite. Guillen and Silvestre proved that solutions of the spatially homogeneous equation never blow up. Bedrossian, Chen, Gualdani, Ji, Vicol and Yang constructed smooth, strictly positive data that develop a finite-time singularity for the inhomogeneous equation with the very hard potentials $|z|^{2+\gamma}$, $\gamma \in (\sqrt{3}, 2]$, in place of the Coulomb kernel.
Problem. Does the local nonnegative classical solution with $f(0) = f_0$ always extend to all $t \ge 0$, continuously at $t = 0$ and smoothly for $t > 0$, with $$\sup_{0 \le t \le T} \|\langle v \rangle^m f(t)\|_{L^\infty_{x,v}} < \infty$$ for every finite $T$ and every $m \ge 0$?
Source: The problem of global-in-time solutions of the inhomogeneous Landau equation for large initial data, as posed in Henderson, Snelson and Tarfulea (2020), §1; stated in this form, for the Coulomb case on the three-torus with smooth, rapidly decaying data, by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, global classical solutions, kinetic equations, Landau equation
References
@article{henderson2020,
author = {Henderson, Christopher and Snelson, Stanley and Tarfulea, Andrei},
title = {Local solutions of the {L}andau equation with rough, slowly decaying initial data},
journal = {Ann. Inst. H. Poincar{\'e} C Anal. Non Lin{\'e}aire},
volume = {37},
number = {6},
year = {2020},
pages = {1345--1377},
doi = {10.1016/j.anihpc.2020.04.004},
eprint = {1909.05914},
archivePrefix = {arXiv},
}
@article{guo2002,
author = {Guo, Yan},
title = {The {L}andau equation in a periodic box},
journal = {Comm. Math. Phys.},
volume = {231},
number = {3},
year = {2002},
pages = {391--434},
doi = {10.1007/s00220-002-0729-9},
}
@article{snelson2026,
author = {Snelson, Stanley and Solomon, Caleb},
title = {A continuation criterion for the {L}andau equation with very soft and {C}oulomb potentials},
journal = {Arch. Ration. Mech. Anal.},
volume = {250},
number = {2},
year = {2026},
pages = {20},
doi = {10.1007/s00205-026-02182-8},
eprint = {2309.15690},
archivePrefix = {arXiv},
}
@misc{golding2026,
author = {Golding, William and Henderson, Christopher and Silvestre, Luis},
title = {Pointwise bounds and obstructions to blowup for the {L}andau and {B}oltzmann equations},
year = {2026},
eprint = {2605.20426},
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{bedrossian2026,
author = {Bedrossian, Jacob and Chen, Jiajie and Gualdani, Maria Pia and Ji, Sehyun and Vicol, Vlad and Yang, Jincheng},
title = {Finite time singularities in the {L}andau equation with very hard potentials},
year = {2026},
eprint = {2602.05981},
archivePrefix = {arXiv},
}
@misc{aim-landau-coulomb-inhomogeneous,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Global classical solutions of the spatially inhomogeneous Landau–Coulomb equation}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/297-landau-coulomb-inhomogeneous.md},
note = {Entry 297 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-00873, "Global classical solutions of the spatially inhomogeneous Landau–Coulomb equation", clopen.solutions.