Research ArticleOpen AccessGoogle Scholar indexed
Existence and Uniqueness of Global Smooth Solutions for Vlasov Maxwell Equations
Department of Mathematics, Jimma University, Jimma, Ethiopia
- 1 Department of Mathematics, Jimma University, Jimma, Ethiopia
Advances in Pure Mathematics·Volume 08 (2018)·Pages 45–76·Published 9 January 2018·DOI10.4236/apm.2018.81005
Copy link · social · email
Abstract
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.
KeywordsMaxwell EquationVlasov EquationExistence and Uniqueness
- 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.
- Glassy, R.T. (1987) Absence of Shocks in an Initially Dilute Collisionless Plasma. Communications in Mathematical Physics, 113, 191-208.
- Calogero, S. (2004) Global Small Solutions of the Vlasov-Maxwell System in the Absence of Incoming Radiation. Indiana University Mathematics Journal, 53, 1331-1363.
- Klainerman, S. and Staffilani, G. (2002) A New Approach to Study the Vlasov-Maxwell System. Communications on Pure and Applied Analysis, 1, 103-125.
- Glass, O. and Han-Kwan, D. (2012) On the Controllability of the Relativistic Vlasov-Maxwell System. arXiv: 1211.7236.
- Rein, G. (1990) Generic Global Solutions of the Relativistic Vlasov-Maxwell System of Plasma Physics. Communications in Mathematical Physics, 135, 41-78.
- Luk, J. and Strain, R.M. (2014) A New Continuation Criterion for the Relativistic Vlasov-Maxwell System. ArXiv e-Prints 1406.0165.
- Evans, L.C. (2010) Partial Differential Equations, Graduate Studies in Mathematics, American Mathematical Society.