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Article Contents

# A toy model for the relativistic Vlasov-Maxwell system

• *Corresponding author: Jonathan Ben-Artzi

The authors thank C. Pallard for explaining to them some of the delicate aspects of the Division Lemma, and the anonymous referees whose comments helped improve the presentation of the paper.
The authors thank Claude Bardos and François Golse who proposed this problem over dinner during the workshop "The Cauchy Problem in Kinetic Theory: Recent Progress in Collisionless Models" which was held at Imperial College London in 2015. That workshop was held in honor of Bob Glassey, to whose memory this paper is dedicated.

The first author acknowledges support from an Engineering and Physical Sciences Research Council Fellowship (EP/N020154/1). The second author acknowledges support from the US National Science Foundation under awards DMS-1911145 and DMS-2107938. The third author acknowledges support from the National Natural Science Foundation of China (11771041, 11831004) and a Marie Sk lodowska-Curie Fellowship (790623)

• The global-in-time existence of classical solutions to the relativistic Vlasov-Maxwell (RVM) system in three space dimensions remains elusive after nearly four decades of mathematical research. In this note, a simplified "toy model" is presented and studied. This toy model retains one crucial aspect of the RVM system: the phase-space evolution of the distribution function is governed by a transport equation whose forcing term satisfies a wave equation with finite speed of propagation.

Mathematics Subject Classification: Primary: 35Q83, 35B40; Secondary: 82D10.

 Citation:

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