Global classical solutions for repulsive relativistic Vlasov–Poisson
The repulsive relativistic Vlasov–Poisson system in three dimensions is $$\partial_t f + \frac{p}{\sqrt{1 + |p|^2}} \cdot \nabla_x f + E \cdot \nabla_p f = 0, \qquad E = -\nabla_x \Phi, \qquad -\Delta_x \Phi = 4\pi \int_{\mathbb{R}^3} f \, dp, \qquad \Phi(x) \to 0 \ (|x| \to \infty),$$ for $f(t, x, p) \ge 0$ on $\mathbb{R}^3_x \times \mathbb{R}^3_p$. For the nonrelativistic system, with $p$ in place of $p / \sqrt{1 + |p|^2}$, Pfaffelmoser and Lions and Perthame proved global existence of classical solutions for general initial data. For the relativistic system, Glassey and Schaeffer proved it for spherically symmetric data with compact support in $x$ and $p$, and for cylindrically symmetric data with compact support whose angular momentum is bounded away from zero; Wang removed the compact support assumptions and replaced the lower bound on the angular momentum by a vanishing-order condition at zero angular momentum.
Problem. For every nonnegative $f_0 \in C_c^\infty(\mathbb{R}^3_x \times \mathbb{R}^3_p)$, does the system have a unique global classical solution with $f(0) = f_0$? More precisely, must the momentum support of the classical solution remain bounded on each finite time interval? No symmetry or smallness assumption is made.
Source: Stated by AIM — Open Applied Problems, after Wang (2023), §1, who asks whether the analogue of Lions and Perthame's global existence theorem holds for the three-dimensional relativistic Vlasov–Poisson system in the plasma physics case; see Glassey and Schaeffer (1985).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, collisionless plasmas, Vlasov–Poisson systems
References
@article{wang2023,
author = {Wang, Xuecheng},
title = {Global solution of the {3D} relativistic {V}lasov--{P}oisson system for a class of large data},
journal = {J. Stat. Phys.},
volume = {190},
number = {10},
year = {2023},
pages = {Paper No. 162},
doi = {10.1007/s10955-023-03178-4},
eprint = {2003.14191},
archivePrefix = {arXiv},
}
@article{glassey1985,
author = {Glassey, Robert T. and Schaeffer, Jack},
title = {On symmetric solutions of the relativistic {V}lasov--{P}oisson system},
journal = {Comm. Math. Phys.},
volume = {101},
number = {4},
year = {1985},
pages = {459--473},
doi = {10.1007/BF01210740},
}
@article{glassey2001,
author = {Glassey, Robert T. and Schaeffer, Jack},
title = {On global symmetric solutions to the relativistic {V}lasov--{P}oisson equation in three space dimensions},
journal = {Math. Methods Appl. Sci.},
volume = {24},
number = {3},
year = {2001},
pages = {143--157},
doi = {10.1002/1099-1476(200102)24:3<143::AID-MMA202>3.0.CO;2-C},
}
@article{pfaffelmoser1992,
author = {Pfaffelmoser, K.},
title = {Global classical solutions of the {V}lasov--{P}oisson system in three dimensions for general initial data},
journal = {J. Differential Equations},
volume = {95},
number = {2},
year = {1992},
pages = {281--303},
doi = {10.1016/0022-0396(92)90033-J},
}
@article{lions1991,
author = {Lions, P.-L. and Perthame, B.},
title = {Propagation of moments and regularity for the 3-dimensional {V}lasov--{P}oisson system},
journal = {Invent. Math.},
volume = {105},
number = {1},
year = {1991},
pages = {415--430},
doi = {10.1007/BF01232273},
}
@misc{aim-relativistic-vlasov-poisson-global,
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 for repulsive relativistic Vlasov–Poisson}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/362-relativistic-vlasov-poisson-global.md},
note = {Entry 362 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-00636, "Global classical solutions for repulsive relativistic Vlasov–Poisson", clopen.solutions.