Large-data global classical solutions of relativistic Vlasov–Maxwell
For a momentum $v \in \mathbb{R}^3$ write $\widehat v = v/\sqrt{1 + |v|^2}$. The one-species relativistic Vlasov–Maxwell system, with the particle mass and charge and the speed of light normalized to $1$, is $$\partial_t f + \widehat v \cdot \nabla_x f + (E + \widehat v \times B) \cdot \nabla_v f = 0,$$ $$\partial_t E = \nabla \times B - j, \qquad \partial_t B = -\nabla \times E, \qquad j = \int_{\mathbb{R}^3} \widehat v f \, dv,$$ for a particle density $f(t, x, v) \ge 0$ on $\mathbb{R}^3_x \times \mathbb{R}^3_v$ and fields $E(t, x)$, $B(t, x)$, with the constraints $\nabla \cdot E = \int f \, dv$ and $\nabla \cdot B = 0$. Smooth compatible data generate a classical solution locally in time, and DiPerna and Lions constructed global weak solutions. Glassey and Strauss proved that a classical solution remains $C^1$ as long as the momentum support of $f$ stays bounded, and proved global classical existence for initially dilute plasmas; Luk and Strain showed that it suffices to bound the momentum support after projection onto any two-dimensional plane.
Problem. Let $f_0 \in C_c^\infty(\mathbb{R}^3_x \times \mathbb{R}^3_v)$ be nonnegative, and let $E_0, B_0$ be smooth electromagnetic data of finite energy with bounded derivatives, satisfying $\nabla \cdot E_0 = \int f_0 \, dv$ and $\nabla \cdot B_0 = 0$. Does the system have a classical solution with these data for all $t \ge 0$ that keeps both divergence constraints and finite field energy, and whose particle momenta stay in a bounded set on every finite time interval? No smallness or symmetry assumption is made on the data.
Source: A long-standing problem for the relativistic Vlasov–Maxwell system (Luk and Strain (2014), abstract; Han-Kwan, Nguyen and Rousset (2025), Introduction). 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, global classical solutions, kinetic theory, Vlasov–Maxwell system
References
@article{glassey1986,
author = {Glassey, Robert T. and Strauss, Walter A.},
title = {Singularity formation in a collisionless plasma could occur only at high velocities},
journal = {Arch. Rational Mech. Anal.},
volume = {92},
number = {1},
year = {1986},
pages = {59--90},
doi = {10.1007/BF00250732},
}
@article{glassey1987,
author = {Glassey, Robert T. and Strauss, Walter A.},
title = {Absence of shocks in an initially dilute collisionless plasma},
journal = {Comm. Math. Phys.},
volume = {113},
number = {2},
year = {1987},
pages = {191--208},
doi = {10.1007/BF01223511},
}
@article{diperna1989,
author = {DiPerna, R. J. and Lions, P.-L.},
title = {Global weak solutions of {V}lasov--{M}axwell systems},
journal = {Comm. Pure Appl. Math.},
volume = {42},
number = {6},
year = {1989},
pages = {729--757},
doi = {10.1002/cpa.3160420603},
}
@article{luk2014,
author = {Luk, Jonathan and Strain, Robert M.},
title = {A new continuation criterion for the relativistic {V}lasov--{M}axwell system},
journal = {Comm. Math. Phys.},
volume = {331},
number = {3},
year = {2014},
pages = {1005--1027},
doi = {10.1007/s00220-014-2108-8},
eprint = {1406.0165},
archivePrefix = {arXiv},
}
@article{hankwan2025,
author = {Han-Kwan, Daniel and Nguyen, Toan T. and Rousset, Fr{\'e}d{\'e}ric},
title = {Linear {L}andau damping for the {V}lasov--{M}axwell system in $\mathbb{R}^3$},
journal = {Ann. PDE},
volume = {11},
number = {2},
year = {2025},
pages = {26},
doi = {10.1007/s40818-025-00217-z},
eprint = {2402.11402},
archivePrefix = {arXiv},
}
@misc{aim-vlasov-maxwell-global-classical,
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 classical solutions of relativistic Vlasov–Maxwell}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/073-vlasov-maxwell-global-classical.md},
note = {Entry 073 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
We prove global existence and uniqueness for arbitrary smooth admissible initial data in the three-dimensional, one-species relativistic Vlasov--Maxwell system. The particle density is initially compactly supported, and the electromagnetic fields have finite energy and bounded derivatives of all orders. The solution remains smooth on every finite time interval. This resolves the large-data global classical regularity problem for this model, without …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: math @ 1581d4c, build/main.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00737, "Large-data global classical solutions of relativistic Vlasov–Maxwell", clopen.solutions.