Result 362, Partial differential equations

Global smoothness for relativistic Vlasov–Maxwell

Proves large-data global existence and uniqueness for the three-dimensional, one-species relativistic Vlasov–Maxwell system. Smooth admissible initial data may be arbitrary provided the particle density is compactly supported and the electromagnetic fields have finite energy and bounded derivatives of every order; the solution remains smooth on every finite time interval.

Lean formalization Reasoning summary (PDF) Proof

The bigger picture

Why it matters

Can a smooth cloud of charged particles develop a mathematical singularity through its own electromagnetic field? This manuscript claims that, for a specified three-dimensional relativistic model, smooth evolution can continue uniquely for all time.

What changes?

The model couples a particle density, describing how particles are distributed in position and momentum, to electric and magnetic fields. For one particle species in three spatial dimensions, the unreviewed manuscript reports global existence and uniqueness for arbitrary smooth admissible initial data. The density must initially have compact support, meaning it vanishes outside a bounded region, and the fields must have finite energy and bounded derivatives of every order. No smallness or symmetry assumption is imposed.

What does that help mathematicians do?

If correct, the result rules out finite-time loss of smoothness for this entire class of initial data, even when the data are large. Uniqueness also excludes competing smooth evolutions from the same starting state. Researchers could therefore study long-time behavior without first assuming that a smooth solution survives. Smoothness on every finite interval does not, however, establish bounds uniform over all time.

Are there practical applications?

Its immediate value is foundational for the mathematics of relativistic charged-particle dynamics: it would establish that this single-species model gives a unique smooth evolution under the stated assumptions. That provides a basis for further analysis of the model, but the reported result does not itself supply numerical error guarantees or extend to systems with multiple particle species.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Global classical solutions of the three-dimensional relativistic Vlasov–Maxwell system

September 23, 2026 49 pages Main result formalized in Lean

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 size or symmetry restrictions on the data.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Global-classical-solutions-of-the-three-dimensional-relativistic-Vlasov-Maxwell-system-September-23-2026,
  author = {{OpenAI}},
  title = {{Global classical solutions of the three-dimensional relativistic Vlasov--Maxwell system}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Global-classical-solutions-of-the-three-dimensional-relativistic-Vlasov-Maxwell-system-September-23-2026/paper.pdf}{OAI:Global-classical-solutions-of-the-three-dimensional-relativistic-Vlasov-Maxwell-system-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/362.md.

Global smoothness for relativistic Vlasov–Maxwell

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalized result gives global existence and uniqueness for the three-dimensional one-species relativistic Vlasov–Maxwell system. The initial particle density is nonnegative, smooth, and compactly supported; the initial fields have bounded derivatives of all orders, finite energy, and satisfy both Gauss constraints. The solution is smooth on every finite time interval, has compact particle phase support there, and is unique among classical solutions. No smallness, symmetry, or neutrality assumption is imposed.

Comparator links

Result Comparator statement
Global classical Vlasov–Maxwell solutions VlasovMaxwell.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.