Bostan, M.; Poupaud, F. Periodic solutions of the 1D Vlasov-Maxwell system with boundary conditions. (English) Zbl 0965.35010 Math. Methods Appl. Sci. 23, No. 14, 1195-1221 (2000). The authors study the 1D Vlasov-Maxwell system with time-periodic boundary conditions in its classical and relativistic form. The main goal of the paper is to prove existence result for the weak periodic solution. The main mathematical difficulty consists of establishing \(L^\infty\)-estimates for the charge and current densities. Then the authors construct the regularized system, which leads to a sequence of approximate solutions. The existence proof uses the Schauder fixed point theorem for the modified problem. Then the passage to the limit in the regularization parameter leads to the main result. The last section treats the relativistic case. Reviewer: Nikita E.Ratanov (Chelyabinsk) Cited in 2 Documents MSC: 35B10 Periodic solutions to PDEs 82D10 Statistical mechanical studies of plasmas Keywords:kinetic theory; time-periodic boundary conditions; weak solution; mild solution; Schauder fixed point theorem; classical and relativistic form; \(L^\infty\)-estimates for the charge and current densities; regularized system PDF BibTeX XML Cite \textit{M. Bostan} and \textit{F. Poupaud}, Math. Methods Appl. Sci. 23, No. 14, 1195--1221 (2000; Zbl 0965.35010) Full Text: DOI References: [1] Arsenev, USSR Comput. Math. Math. Phys. 15 pp 131– (1975) · doi:10.1016/0041-5553(75)90141-X [2] Asano, Stud. Math. Appl. 18 pp 369– (1986) [3] Bardos, Ann. Sci. Ecole Norm. Sup. 4 pp 185– (1969) [4] Weak solutions of the initial-boundary value problem for the Vlasov-Poisson system. Preprint. [5] Boundary value problems for the Vlasov-Maxwell system. Semin. Equations Deriv. Partielles, Ec. Polytech., Cent. Math., Palaiseau Semi 1992-1993, Expose No. 4, 1993. [6] Modélisation et simulation numérique du régime de Child-Langmuir. Thèse de l’Ecole Polytechnique, Palaiseau, 1995. [7] Bostan, C.R. Acad. Sci. Paris t. 325 pp 1333– (1997) · Zbl 0895.35079 · doi:10.1016/S0764-4442(97)82365-8 [8] Periodic solutions of the Vlasov-Poisson system with boundary conditions. Report INRIA-CERMICS, No. 3518, October 1998. [9] Controllability methods for the calculation of periodic solutions of the Vlasov-Maxwell system with boundary conditions. Report INRIA-CERMICS, No. 3534, October 1998. [10] Diperna, Comm. Pure Appl. Math XVII pp 729– (1989) · Zbl 0698.35128 · doi:10.1002/cpa.3160420603 [11] Diperna, Invent. Math 98 pp 511– (1989) · Zbl 0696.34049 · doi:10.1007/BF01393835 [12] Degond, Ann. Sci. Ecole Norm. Sup. Ser. IV 19 pp 519– (1986) [13] Degond, Math. Meth. in the Appl. Sci. 8 pp 533– (1986) · Zbl 0619.35088 · doi:10.1002/mma.1670080135 [14] Regularité de la solution des équations cinétiques en physiques de plasmas. Semin. Equations Deriv. Partielles 1985-86, Expose No. 18, 1986. [15] Greengard, Comm. Pure Appl. Math. XLIII pp 473– (1990) · Zbl 0721.35084 · doi:10.1002/cpa.3160430404 [16] Guo, Comm. Math. Phys. 154 pp 245– (1993) · Zbl 0787.35072 · doi:10.1007/BF02096997 [17] Guo, Indiana Univ. Math. J. 43 pp 255– (1994) · Zbl 0799.35031 · doi:10.1512/iumj.1994.43.43013 [18] Pfaffelmoser, J. Differential Equations 95 pp 281– (1992) · Zbl 0810.35089 · doi:10.1016/0022-0396(92)90033-J [19] Poupaud, Forum Math. 4 pp 499– (1992) · Zbl 0785.35020 · doi:10.1515/form.1992.4.499 [20] Lions, Invent. Math. 105 pp 415– (1991) · Zbl 0741.35061 · doi:10.1007/BF01232273 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.