Abstract
We consider a space-periodic version of the relativistic Vlasov-Maxwell system describing a collisionless plasma consisting of electrons and positively charged ions. As our main result, we prove that certain spacially homogeneous stationary solutions are nonlinearly stable. To this end we also establish global existence of weak solutions to the corresponding initial value problem. Our investigation is motivated by a corresponding result for the Vlasov-Poisson system, cf. [1, 14].
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References
Batt, J. & Rein, G., A rigorous stability result for the Vlasov-Poisson system in three dimensions, to appear in Anal. di Mat. Pura ed Appl.
Batt, J. & Rein, G., Global classical solutions of the periodic Vlasov-Poisson system in three dimensions, C. R. Acad. Sci. Paris 313, 411–416 (1991).
DiPerna, R. J. & Lions, P. L., Global weak solutions of Vlasov-Maxwell systems, Commun. Pure Appl. Math. 42, 729–757 (1989).
Glassey, R. & Schaeffer, J., Global existence for the relativistic Vlasov-Maxwell system with nearly neutral initial data, Commun. Math. Phys. 119, 353–384 (1988).
Glassey, R. & Strauss, W., Absence of shocks in an initially dilute collisionless plasma, Commun. Math. Phys. 113, 191–208 (1987).
Holm, D. D., Marsden, J. E., Ratiu, T.. & Weinstein, A., Nonlinear stability of fluid and plasma equilibria, Physics Reports 123, 1–116 (1985).
Horst, E., Global solutions of the relativistic Vlasov-Maxwell system of plasma physics, Habilitationsschrift, Universität München, 1986.
Horst, E., On the asymptotic growth of the solutions of the Vlasov-Poisson system, preprint, 1991.
Kruse, K.-O., Ein neuer Zugang zur globalen Existenz von Distributionenlösungen des Vlasov-Maxwell-Systems partieller Differentialgleichungen, Diplomarbeit, Universität München, 1991.
Leis, R., Initial Boundary Value Problems in Mathematical Physics, Stuttgart, 1986.
Lions, P. L. & Perthame, B., Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Invent, math. 105, 415–430 (1991).
Marchioro, C. & Pagani, E., Nonlinear stability of a spatially symmetric solution of the relativistic Poisson-Vlasov equation, Rend. Sem. Mat. Univ. Padova 78, 125–143 (1987).
Marchioro, C. & Pulvirenti, M., Some considerations on the nonlinear stability of stationary Euler flows, Commun. Math. Phys. 100, 343–354 (1985).
Marchioro, C. & Pulvirenti, M., A note on the nonlinear stability of a spatially symmetric Vlasov-Poisson flow, Math. Meth. Appl. Sci. 8, 284–288 (1986).
Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, New York, 1983.
Pfaffelmoser, K., Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Diff. Eqns. 95, 281–303 (1992).
Reed, M. & Simon, B., Methods of Modern Mathematical Physics, Vol. II, New York, 1975.
Rein, G., Generic global solutions of the relativistic Vlasov-Maxwell system of plasma physics, Commun. Math. Phys. 135, 41–78 (1990).
Schaeffer, J., Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Commun. Partial Differential Equs. 16, 1313–1335 (1991).
Wan, Y.-H., Nonlinear stability of stationary spherically symmetric models in stellar dynamics, Arch. Rational Mech. Anal. 112, 83–95 (1990).
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Communicated by K. Kirchgässner
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Kruse, K.O., Rein, G. A stability result for the relativistic Vlasov-Maxwell system. Arch. Rational Mech. Anal. 121, 187–203 (1992). https://doi.org/10.1007/BF00375417
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DOI: https://doi.org/10.1007/BF00375417