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On local solutions of the initial value problem for the Vlasov-Maxwell equation. (English) Zbl 0631.76090
The initial value problem of the Vlasov-Maxwell equation has a unique solution in a time interval [0,T] for each initial data in some function space. T is estimated by the size of the initial data. The solution is classical, if the initial data is smooth.

MSC:
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
35Q99 Partial differential equations of mathematical physics and other areas of application
82B40 Kinetic theory of gases in equilibrium statistical mechanics
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
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[1] Arsen’ev, A. A.: Local uniqueness and existence of a classical solution of Vlasov system of equation. Sov. Math., Dokl.15, 1223-1225 (1974) · Zbl 0311.76036
[2] Bardos, C., Degond, P.: Global existence for the Vlasov-Poisson Equation in 3 space variables with small initial data. Ann. Inst. Henri Poincar?-Analyse non lin?aire2, 101-118 (1985) · Zbl 0593.35076
[3] Batt, J.: Global symmetric solutions of the initial value problem of steller dynamics. J. Differ. Equations25, 342-364 (1977) · Zbl 0366.35020 · doi:10.1016/0022-0396(77)90049-3
[4] Cooper. J., Klimas, A.: Boundary value problems for the Vlasov-Maxwell equation in one dimension. J. Math. Anal. Appl.75, 306-329 (1980) · Zbl 0454.35075 · doi:10.1016/0022-247X(80)90082-7
[5] Duniec, J.: On an initial value problem for nonlinear systems of Vlasov-Maxwell equations. Bull. Acad. Plar. Sci. Ser. Sci. Tech.21, 97-102 (1973) · Zbl 0254.35015
[6] Iordanskii, S. V.: The Cauchy problem for the kinetic equation of plasma. Mat. Inst. Steklova60, 181-194 (1961) English transl. Transl. Am. Math. Soc. (2)35, 351-363 (1964)
[7] Neunzert, H., Petry, K. H.: Ein Existenzsatz fur die Vlasov-Gleichung mit selbstkonsistentem Magnetfeld. Math. Mech. Appl. Sci.2, 429-444 (1980) · Zbl 0454.35076 · doi:10.1002/mma.1670020406
[8] Ukai, S., Okabe, T.: On classical solutions in the large in time of two dimensional Vlasov’s equation. Osaka J. Math.15, 245-261 (1978) · Zbl 0405.35002
[9] Wollmann, S.: An existence and uniqueness theorem for the Vlasov-Maxwell system. Commun. Pure Appl. Math.37, 457-462 (1984) · Zbl 0592.45010 · doi:10.1002/cpa.3160370404
[10] Degond, P.: Local existence of solutions of the Vlasov-Maxwell equations and convergence to the Vlasov-Poisson for infinite light velocity (1984), Preprint · Zbl 0619.35088
[11] Asano, K. Ukai, S.: On the Vlasov-Poisson limit of the Vlasov-Maxwell equations (1985), to appear in: Qualitative analysis of nonlinear differential equations. Studies in Math. its Appl., North-Holland · Zbl 0623.35059
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