Vlasov–Maxwell regularity

The problem asks whether smooth admissible data for the three-dimensional relativistic Vlasov–Maxwell system always produce global regular solutions, or whether breakdown can occur. Published sources continue to describe the unrestricted question as open.

Known results

• Glassey–Strauss and Schaeffer proved global existence in small-data regimes, with later work removing some support assumptions.
• Global existence is known for sufficiently small smooth data with decay, including scattering results.
• Arbitrarily large global solutions are known under spherical symmetry and for some cylindrically symmetric data.
• Continuation criteria give conditional global regularity when suitable momentum-integrability or support bounds persist.

Community submission (unverified), 2026-10-07

A submitted claim alleges that an attached manuscript proves global existence and uniqueness for arbitrary smooth admissible data in the one-species relativistic system. No independently retrieved source assesses this claim, so it does not establish a solution.

Current status (as of October 2026): Restricted-data and symmetry results are established, but global regularity or breakdown for general smooth three-dimensional data remains open; the submitted complete-solution claim is unverified.