Communications in Mathematical Sciences

Volume 18 (2020)

Number 1

The relativistic Vlasov–Maxwell equations for strongly magnetized plasmas

Pages: 123 – 162



Christophe Cheverry (Institut Mathématique de Rennes, France)

Slim Ibrahim (Department of Mathematics and Statistics, University of Victoria, British Columbia, Canada)


An important challenge in plasma physics is to determine whether ionized gases can be confined by strong magnetic fields. After properly formulating the model, this question leads to a penalized version of the relativistic Vlasov–Maxwell system, marked by the role of a singular factor $\varepsilon^{-1}$ corresponding to the inverse of a cyclotron frequency. In this paper, we prove in this context the existence of classical $\mathcal{C}^1$-solutions for a time independent of $\varepsilon$. We also investigate the stability of these smooth solutions.


kinetic equations, Vlasov–Maxwell system , magnetized plasmas, lifespan of classical solutions, momentum support condition

2010 Mathematics Subject Classification

35Q60, 35Q61, 35Q83, 82D10, 92C37

A part of this work was done while C. Cheverry was visiting the Department of Mathematics and Statistics of the University of Victoria, and S. Ibrahim was visiting IRMAR, the “Institut de Recherche Mathématique de Rennes”. They both thank all members and staff at the two institutions for their warm hospitality. They also thank D. Preissl for interesting discussions about the text. Both C. Cheverry and S. Ibrahim were supported by France-Canada Research Fund. S. Ibrahim was supported by NSERC grant (371637-2014).

Received 4 April 2019

Accepted 3 September 2019

Published 1 April 2020