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Local existence of solutions of the Vlasov-Maxwell equations and convergence to the Vlasov-Poisson equations for infinite light velocity. (English) Zbl 0619.35088
We prove the local existence of smooth solutions for the Vlasov-Maxwell equations in three space variables. The existence time for such solutions is independant of the light velocity c. Then we derive regularity results for both the Vlasov-Poisson and the Vlasov-Maxwell equations. The last part of the paper is devoted to a proof of weak and strong convergence of the Vlasov-Maxwell equations towards the Vlasov-Poisson equations, when the light velocity c goes to infinity.

35Q99 Partial differential equations of mathematical physics and other areas of application
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
Full Text: DOI
[1] Arsenev, Global existence of a weak solution of Vlasov’s system of equations, USSR comput. Math. and Math. Phys. 15 pp 131– (1975) · doi:10.1016/0041-5553(75)90141-X
[2] Iordanskit, The Cauchy problem for the kinetic equation of Plasma, Amer. Math. Soc. Trans. Ser. 2- 35 pp 351– (1964) · Zbl 0127.21902 · doi:10.1090/trans2/035/12
[3] Ukai, On the classical solution in the large in time of the two dimensional Vlasov equations. Osaka, J. of Math. 15 pp 245– (1978)
[4] Horst, On the classical solutions of the initial value problem for the unmodified nonlinear Vlasov equation, Math. Meth in the Appl. Sci. 3 pp 229– (1981) · Zbl 0463.35071 · doi:10.1002/mma.1670030117
[5] Illner, An existence theorem for the unmodified Vlasov Equation, Math. Meth. in the Appl. Sci. 1 pp 530– (1979) · Zbl 0415.35076 · doi:10.1002/mma.1670010410
[6] Bardos , C. Degond , P. Centre de Maths. Appliquées Ecole Polytechnique Palaiseau (France)
[7] Wollman , S. Existence and uniqueness theory of The Vlasov equation. Internal report; Courant Institute of Math. Science, New-York (oct. 82) · Zbl 0645.35013
[8] Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rat. Mech. Anal. 58 pp 181– (1975) · Zbl 0343.35056 · doi:10.1007/BF00280740
[9] Klainerman, Compressible and incompressible fluids, C. P. A. M. 35 pp 629– (1982)
[10] Added , S. Equations du plasma de Langmuir et équation de schrödinger non linéaire, régularité et approximation.
[11] Klainerman, Global existence for non linear wave equations, C. P. A. M. 33 pp 43– (1980) · Zbl 0405.35056
[12] Klainerman, Global small amplitude solutions to non linear evolution equations, C. P. A. M. · Zbl 0509.35009
[13] Asano , K. On local solutions of the initial value problems for the Vlasov-Maxwell equation. · Zbl 0631.76090
[14] Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires (1969)
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