Global classical solutions of the 3D relativistic Vlasov-Maxwell system for large data
The relativistic Vlasov-Maxwell system couples a particle density on , transported by with , to Maxwell's equations with charge and current densities computed from . Wollman proved local existence of classical solutions, DiPerna and Lions global weak solutions, and Glassey and Strauss showed that a classical solution continues as long as its momentum support stays bounded and proved global existence for small (dilute) data. Global classical existence was known only under smallness, symmetry or reduced dimension. For arbitrary smooth, compactly supported particle data and smooth finite-energy fields satisfying the constraints, does the classical solution exist for all time?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Kinetic theory, nonlinear PDE
- Posed by
- Classical open problem framed by R. Glassey and W. Strauss (1986 continuation criterion); no single poser named in the manuscript
- Year posed
- —
- Years open
- —
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Lean-checked, statement unaudited
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 55 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Claims that every smooth admissible datum (, , fields in satisfying , ) generates a unique global classical solution of the one-species relativistic Vlasov-Maxwell system in three dimensions, smooth on every with compact phase-space support there. The key estimate controls the signed momentum impulse along characteristics, integrating the retarded force along source trajectories before taking absolute values, which rules out momentum blow-up and lets the Glassey-Strauss criterion apply. It treats one species without background charge; it does not give uniform-in-time momentum bounds (the support may grow with ) or asymptotic behaviour, and it does not cover the non-relativistic or multi-species systems explicitly.
What the AI did
The release README says the manuscripts were produced by an unreleased internal OpenAI model, which was posed some 4,000 open research problems during an evaluation, with the vast majority of results obtained by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute each. This manuscript is not among the README's named exceptions. It is credited to OpenAI with no human author named.
Verification
No independent mathematician has checked this yet. Theorem 1.1 was read against the posed problem and claims global existence, uniqueness and smoothness on every finite interval with no size or symmetry restriction. formalization.yaml lists OAI.RVM.global_classical_solution as the main result. Its 112-line comparator statement defines the system itself from Mathlib primitives (fderiv, Bochner integrals, Lp): admissible data are smooth, compactly supported, nonnegative f0 and smooth bounded-derivative L2 fields satisfying both Gauss constraints; the conclusion is a C1 classical solution with L2-continuous fields, compact phase support on finite horizons, smoothness on every slab, and uniqueness among classical solutions. That states the headline claim; the hand-written definitions were not audited against the paper here. Not rebuilt here. Scope the paper states: one species, no background charge; the proof uses the Glassey-Strauss criterion as stated by Luk and Strain.