Global weak solutions of Vlasov-Maxwell systems. (English) Zbl 0698.35128

Using the method they developed in a previous paper to show global existence for the Cauchy problem for the Boltzmann equation, the authors are able to prove the existence of global weak solutions to the Vlasov- Maxwell system with no restriction on the size of initial data. The proof is basically in two steps. Firstly a weak stability theorem is demonstrated, based on a series of complex estimates. Next, global existence is established as an application of the previous result, going through a regularization procedure. The question of uniqueness is left open.
In the last section the authors mention several possible extensions of their theory, concerning different boundary value problems or various modifications of the governing differential equations.
Reviewer: A.Fasano


35Q99 Partial differential equations of mathematical physics and other areas of application
35D05 Existence of generalized solutions of PDE (MSC2000)
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[1] Arsenev, U.S.S.R. Comp. Math. and Math. Phys. 15 pp 131– (1975)
[2] Batt, J. Diff. Eq. 25 pp 342– (1977)
[3] Batt, Ann. of Nucl. Energy 7 pp 213– (1980)
[4] and , Interpolation Spaces, Springer-Verlag, Berlin, 1976. · doi:10.1007/978-3-642-66451-9
[5] Cooper, J. Math. Anal. Appl. 75 pp 306– (1980)
[6] Régularité de la solution des équations cinétiques en physique des plasmas, Séminaire EDP, 1985–1986, Ecole Polytechnique, Palaiseau.
[7] DiPerna, Comm. Math. Phys. (1989)
[8] DiPerna, Ann. Math. (1989)
[9] C.R. Acad. Sci. Paris 306 pp 343– (1988)
[10] DiPerna, See also C. R. Acad. Sci. Paris 307 pp 655– (1988)
[11] Duniec, Bull. Acad. Polon. Sci. Ser. Sci. Tech. 21 pp 97– (1973)
[12] Moyennes de solutions d’équations aux dérivées partidles, Séminaire, EDP, 1986–1987, Ecole Polytechnique, Palaiseau.
[13] personal communication.
[14] Glassey, Comm. Math. Phys.
[15] Glassey, Contemporary Mathematics 28 pp 269– (1984) · Zbl 0556.35122 · doi:10.1090/conm/028/751989
[16] Golse, J. Funct. Anal. 76 pp 110– (1988)
[17] Golse, C.R. Acad. Sci. Paris 301 pp 341– (1985)
[18] Global solutions of the relativistic Vlasov-Maxwell system of plasma physics, Habilitationsschrift, Univ. München, 1986.
[19] Neunzert, Math. Math. in the Appl. Sci. 2 pp 429– (1980)
[20] Wollman, Comm. Pure Appl. Mat. 37 pp 457– (1984)
[21] Illner, Math. Meth. in the Appl. Sci. 1 pp 530– (1979)
[22] Glassey, Arch. Rat. Mech Anal. 92 pp 59– (1986)
[23] Glassey, Comm. Math. Phys. 113 pp 191– (1987)
[24] and , Global existence for the relativistic Vlasov-Maxwell system with nearly neutral initial data, preprint. · Zbl 0673.35070
[25] Bardos, Ann. Inst. Henri Poin., Anal. Nonlin. 2 pp 101– (1985)
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