A new continuation criterion for the relativistic Vlasov-Maxwell system

TL;DR

This paper introduces a new criterion for the global regularity of solutions to the relativistic Vlasov-Maxwell system, requiring boundedness of the momentum support projection onto any two-dimensional plane, advancing understanding of solution continuation.

Contribution

It establishes that boundedness of the momentum support's projection onto any two-dimensional plane suffices for solution regularity, improving previous criteria that required full support boundedness.

Findings

Solution remains $C^1$ if the projected momentum support is bounded.

Reduces the regularity criterion to two-dimensional projections.

Provides a new continuation criterion for the relativistic Vlasov-Maxwell system.

Abstract

The global existence of solutions to the relativistic Vlasov-Maxwell system given sufficiently regular finite energy initial data is a longstanding open problem. The main result of Glassey-Strauss (1986) shows that a solution $(f, E, B)$ remains $C^1$ as long as the momentum support of $f$ remains bounded. Alternate proofs were later given by Bouchut-Golse-Pallard (2003) and Klainerman-Staffilani (2002). We show that only the boundedness of the momentum support of $f$ after projecting to any two dimensional plane is needed for $(f, E, B)$ to remain $C^1$.