×

zbMATH — the first resource for mathematics

The Vlasov-Maxwell-Fokker-Planck system in two space dimensions. (English) Zbl 1336.35227
Summary: We consider the Cauchy problem for the Vlasov-Maxwell-Fokker-Planck system in the plane. It is shown that for smooth initial data, as long as the electromagnetic fields remain bounded, then their derivatives do also. Glassey and Strauss have shown this to hold for the relativistic Vlasov-Maxwell system in three dimensions, but the method here is totally different. In the work of Glassey and Strauss, the relativistic nature of the particle transport played an essential role. In this work, the transport is nonrelativistic, and smoothing from the Fokker-Planck operator is exploited.
MSC:
35L60 First-order nonlinear hyperbolic equations
35Q83 Vlasov equations
82C22 Interacting particle systems in time-dependent statistical mechanics
82D10 Statistical mechanics of plasmas
35Q61 Maxwell equations
35Q84 Fokker-Planck equations
PDF BibTeX Cite
Full Text: DOI
References:
[1] Glassey, Singularity formation in a collisionless plasma could occur only at high velocities, Archive for Rational Mechanics and Analysis 92 (1) pp 59– (1986) · Zbl 0595.35072
[2] Victory, On classical solutions of Vlasov-Poisson-Fokker-Planck systems, Indiana University Mathematics Journal 39 (1) pp 105– (1990) · Zbl 0674.60097
[3] DiPerna, Global weak solutions of Vlasov-Maxwell systems, Communications on Pure and Applied Mathematics 42 (6) pp 729– (1989) · Zbl 0698.35128
[4] Glassey, The ”two and one-half dimensional” relativistic Vlasov-Maxwell system, Communications in Mathematical Physics 185 pp 257– (1997) · Zbl 0883.35118
[5] Glassey, The relativistic Vlasov-Maxwell system in two space dimensions: parts I and II, Archive for Rational Mechanics and Analysis 141 pp 331– (1998) · Zbl 0907.76100
[6] Lions, Propagation of moments and regularity for the 3-dimensional Vlasov-Pisson system, Inventiones Mathematicae 105 (2) pp 415– (1991) · Zbl 0741.35061
[7] Pankavich, Global existence for the Vlasov-Poisson system with steady spatial asymptotics, Communications in Partial Differential Equations 31 (1-3) pp 349– (2006) · Zbl 1100.35066
[8] Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, Journal of Differential Equations 95 (2) pp 281– (1992) · Zbl 0810.35089
[9] Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Communications in Partial Differential Equations 16 (8-9) pp 1313– (1991) · Zbl 0746.35050
[10] Pankavich, Global classical solutions of the ”one and one-half dimensional” Vlasov-Maxwell-Fokker-Planck System, Communications in Mathematical Sciences pp (to appear)– · Zbl 1332.35214
[11] Chae, The global classical solution of the Vlasov-Maxwell-Fokker-Planck system near Maxwellian, Mathematical Models and Methods in Applied Sciences 21 (5) pp 1007– (2011) · Zbl 1244.35146
[12] Yang, Global classical solutions for the Vlasov-Maxwell-Fokker-Planck system, SIAM Journal on Mathematical Analysis 42 (1) pp 459– (2010) · Zbl 1219.35302
[13] Lai, On the one- and one-half dimensional relativistic Vlasov-Fokker-Planck-Maxwell system, Mathematical Methods in the Applied Sciences 18 (13) pp 1013– (1995) · Zbl 0839.45009
[14] Lai, One the one-and-one-half-dimensional relativistic Vlasov-Maxwell-Fokker-Planck system with non-vanishing viscosity, Mathematical Methods in the Applied Sciences 21 (14) pp 1287– (1998) · Zbl 0911.35091
[15] Bouchut, Existence and uniqueness of a global smooth solution for the Vlasov-Poisson-Fokker-Planck system in three dimensions, Journal of Functional Analysis 111 (1) pp 239– (1993) · Zbl 0777.35059
[16] Degond, Global existence of smooth solutions for the Vlasov-Fokker-Planck equation in 1 and 2 space dimensions, Annales Scientifiques de l’Ecole Normal Superieure (4) 19 (4) pp 519– (1986) · Zbl 0619.35087
[17] Glassey, The Cauchy problem in kinetic theory (1996) · Zbl 0858.76001
[18] Felix, Spatially homogeneous solutions of the Vlasov-Nordstrom-Fokker-Planck system, Journal of Differential Equations 257 (10) pp 3700– (2014) · Zbl 1303.35113
[19] Pankavich, Global classical solutions of the one and one-half dimensional relativistic Vlasov-Maxwell-Fokker-Planck system, Kinetic and Related Models 8 (1) pp 169– (2015) · Zbl 1311.35316
[20] Pankavich, A short proof of increased parabolic regularity, arXiv preprint pp 1502.01773– · Zbl 1322.35048
[21] Batt, Global symmetric solutions of the initial-value problem of stellar dynamics, Journal of Differential Equations 25 pp 342– (1977) · Zbl 0366.35020
[22] Adams, Sobolev Spaces (1975)
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.