Global existence of classical solutions to the
relativistic Vlasov-Maxwell system, given sufficiently regular initial data, is
a long-standing open problem. The aim of this project is to present in details
the results of a paper published in 1986 by Robert Glassey and Walter Strauss.
In that paper, a sufficient condition for the global existence of a smooth
solution to the relativistic Vlasov-Maxwell system is derived. In the following, the resulting theorem is proved by taking initial data , . A small data global existence result is presented
as well.
References
[1]
Glassey, R.T. and Strauss, W.A. (1986) Singularity Formation in a Collisionless Plasma Could Occur Only at a High Velocities. Archive for Rational Mechanics and Analysis, 9, 59-90.
[2]
Glassy, R.T. (1987) Absence of Shocks in an Initially Dilute Collisionless Plasma. Communications in Mathematical Physics, 113, 191-208.
[3]
Calogero, S. (2004) Global Small Solutions of the Vlasov-Maxwell System in the Absence of Incoming Radiation. Indiana University Mathematics Journal, 53, 1331-1363.
[4]
Klainerman, S. and Staffilani, G. (2002) A New Approach to Study the Vlasov-Maxwell System. Communications on Pure and Applied Analysis, 1, 103-125.
[5]
Glass, O. and Han-Kwan, D. (2012) On the Controllability of the Relativistic Vlasov-Maxwell System. arXiv: 1211.7236.
[6]
Rein, G. (1990) Generic Global Solutions of the Relativistic Vlasov-Maxwell System of Plasma Physics. Communications in Mathematical Physics, 135, 41-78.
[7]
Luk, J. and Strain, R.M. (2014) A New Continuation Criterion for the Relativistic Vlasov-Maxwell System. ArXiv e-Prints 1406.0165.
[8]
Evans, L.C. (2010) Partial Differential Equations, Graduate Studies in Mathematics, American Mathematical Society.
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