High-electric-field limit for the Vlasov-Maxwell-Fokker-Planck system
Bostan, Mihai ; Goudon, Thierry
Annales de l'I.H.P. Analyse non linéaire, Tome 25 (2008), p. 1221-1251 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU) 
@article{AIHPC_2008__25_6_1221_0,
     author = {Bostan, Mihai and Goudon, Thierry},
     title = {High-electric-field limit for the Vlasov-Maxwell-Fokker-Planck system},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {25},
     year = {2008},
     pages = {1221-1251},
     doi = {10.1016/j.anihpc.2008.07.004},
     mrnumber = {2466328},
     zbl = {1157.35486},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2008__25_6_1221_0}
}

Bostan, Mihai; Goudon, Thierry. High-electric-field limit for the Vlasov-Maxwell-Fokker-Planck system. Annales de l'I.H.P. Analyse non linéaire, Tome 25 (2008) pp. 1221-1251. doi : 10.1016/j.anihpc.2008.07.004. http://gdmltest.u-ga.fr/item/AIHPC_2008__25_6_1221_0/

[1] Arnold A., Carrillo J.-A., Gamba I., Shu C.-W., Low and high field scaling limits for the Vlasov- and Wigner-Poisson-Fokker-Planck system, Transport Theory Statist. Phys. 30 (2-3) (2001) 121-153. | MR 1848592 | Zbl 1106.82381

[2] Arnold A., Markowich P., Toscani G., Unterreiter A., On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. Partial Differential Equations 26 (1-2) (2001) 43-100. | MR 1842428 | Zbl 0982.35113

[3] Bakry D., Emery M., Hypercontractivité de semi-groupes de diffusion, C. R. Acad. Sci. Paris Sér. I Math. 299 (15) (1984) 775-778. | MR 772092 | Zbl 0563.60068

[4] Bardos C., Golse F., Levermore C.D., Fluid dynamic limits of kinetic equations II. Convergence proofs for the Boltzmann equation, Comm. Pure Appl. Math. XLVI (1993) 667-753. | MR 1213991 | Zbl 0817.76002

[5] Ben Abdallah N., Degond P., Markowich P., Schmeiser C., High field approximation of the spherical harmonics expansion model for semiconductors, Z. Angew. Math. Phys. 52 (2) (2001) 201-230. | MR 1834528 | Zbl 1174.82345

[6] Bers A., Delcroix J.-L., Physique des plasmas, EDP Sciences, 2000.

[7] Berthelin F., Vasseur A., From kinetic equations to multidimensional isentropic gas dynamics before shocks, SIAM J. Math. Anal. 36 (6) (2005) 1807-1835. | MR 2178222 | Zbl 1130.35090

[8] Bostan M., Goudon Th., Low field regime for the relativistic Vlasov-Maxwell-Fokker-Planck system; the one and one half dimensional case, Kinetic Related Models 1 (1) (2008) 139-169. | MR 2383720 | Zbl pre05300207

[9] Bouchut F., Existence and uniqueness of a global smooth solution for the Vlasov-Poisson-Fokker-Planck system in three dimensions, J. Funct. Anal. 111 (1993) 239-258. | MR 1200643 | Zbl 0777.35059

[10] Bouchut F., Smoothing effect for the nonlinear Vlasov-Poisson-Fokker-Planck system, J. Differential Equations 122 (1995) 225-238. | MR 1355890 | Zbl 0840.35053

[11] Bouchut F., Golse F., Pallard C., Classical solutions and the Glassey-Strauss theorem for the 3D Vlasov-Maxwell system, Arch. Ration. Mech. Anal. 170 (1) (2003) 1-15. | MR 2012645 | Zbl 1044.76075

[12] Bourbaki N., Éléments de Mathématiques, Fascicule XXXV, Livre VI, Chapitre IX, Intégration, Hermann, Paris, 1969. | Zbl 0026.38902

[13] Brenier Y., Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations 25 (2000) 737-754. | MR 1748352 | Zbl 0970.35110

[14] Brenier Y., Mauser N., Puel M., Incompressible Euler and e-MHD as scaling limits of the Vlasov-Maxwell system, Commun. Math. Sci. 1 (3) (2003) 437-447. | MR 2069939 | Zbl 1089.35048

[15] Carrillo J.-A., Soler J., On the initial value problem for the Vlasov-Poisson-Fokker-Planck system with initial data in L p spaces, Math. Methods Appl. Sci. 18 (10) (1995) 825-839. | MR 1343393 | Zbl 0829.35096

[16] Carrillo J.-A., Labrunie S., Global solutions for the one-dimensional Vlasov-Maxwell system for laser-plasma interaction, Math. Models Methods Appl. Sci. 16 (1) (2006) 19-57. | MR 2194980 | Zbl 1106.35110

[17] Cercignani C., Gamba I.M., Levermore C.D., High field approximations to a Boltzmann-Poisson system and boundary conditions in a semiconductor, Appl. Math. Lett. 10 (4) (1997) 111-117. | MR 1458163 | Zbl 0894.76072

[18] Chandrasekhar S., Brownian motion, dynamical friction and stellar dynamics, Rev. Mod. Phys. 21 (1949) 383-388. | MR 31822 | Zbl 0036.43003

[19] Csiszar I., Information-type measures of difference of probability distributions and indirect observations, Studia Sci. Math. Hungar. 2 (1967) 299-318. | MR 219345 | Zbl 0157.25802

[20] Degond P., Global existence of smooth solutions for the Vlasov-Fokker-Planck equations in 1 and 2 space dimensions, Ann. Scient. Ecole Normale Sup. 19 (1986) 519-542. | Numdam | MR 875086 | Zbl 0619.35087

[21] Degond P., Jungel A., High field approximation of the energy-transport model for semiconductors with non-parabolic band structure, Z. Angew. Math. Phys. 52 (6) (2001) 1053-1070. | MR 1877692 | Zbl 0991.35043

[22] Diperna R., Lions P.-L., Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math. 42 (6) (1989) 729-757. | MR 1003433 | Zbl 0698.35128

[23] B. Dubroca, R. Duclous, F. Filbet, V. Tikhonchuk, High order resolution of the Maxwell-Fokker-Planck-Landau model intended for ICF/Fast ignition applications, CELIA-Université Bordeaux 1, in preparation.

[24] Glassey R., Strauss W., Singularity formation in a collisionless plasma could occur only at high velocities, Arch. Ration. Mech. Anal. 9 (1) (1986) 59-90. | MR 816621 | Zbl 0595.35072

[25] Golse F., Saint-Raymond L., The Vlasov-Poisson system with strong magnetic field in quasineutral regime, Math. Models Methods Appl. Sci. 13 (5) (2003) 661-714. | MR 1978931 | Zbl 1053.82032

[26] Goudon T., Nieto J., Poupaud F., Soler J., Multidimensional high-field limit of the electrostatic Vlasov-Poisson-Fokker-Planck system, J. Differential Equations 213 (2) (2005) 418-442. | MR 2142374 | Zbl 1072.35176

[27] Goudon T., Hydrodynamic limit for the Vlasov-Poisson-Fokker-Planck system: Analysis of the two-dimensional case, Math. Models Methods Appl. Sci. 15 (5) (2005) 737-752. | MR 2139941 | Zbl 1074.82021

[28] Goudon T., Jabin P.-E., Vasseur A., Hydrodynamic limits for the Vlasov-Navier-Stokes equations. Part II: Fine particles regime, Indiana Univ. Math. J. 53 (2004) 1517-1536. | MR 2106334 | Zbl 1085.35117

[29] Guo Y., The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math. 153 (3) (2003) 593-630. | MR 2000470 | Zbl 1029.82034

[30] V. Grandgirard, Y. Sarrazin, X. Garbet, G. Dif-Pradalier, P. Ghendrih, N. Crouseilles, G. Latu, E. Sonnendrucker, N. Besse, P. Bertrand, GYSELA, a full-f global gyrokinetic semi-Lagrangian code for ITG turbulence simulations, in: Proceedings of Theory of Fusion Plasmas, Varenna, 2006.

[31] Klainerman S., Staffilani G., A new approach to study the Vlasov-Maxwell system, Comm. Pure Appl. Anal. 1 (1) (2002) 103-125. | MR 1877669 | Zbl 1037.35088

[32] Kullback S., A lower bound for discrimination information in terms of variation, IEEE Trans. Inform. Theory 4 (1967) 126-127.

[33] Lai R., On the one and one-half dimensional relativistic Vlasov-Maxwell-Fokker-Planck system with non-vanishing viscosity, Math. Meth. Appl. Sci. 21 (1998) 1287-1296. | MR 1642550 | Zbl 0911.35091

[34] Markowich P., Ringhofer C., Quantum hydrodynamics for semiconductors in the high field case, Appl. Math. Lett. 7 (5) (1994) 37-41. | MR 1350607 | Zbl 0814.35128

[35] Nieto J., Poupaud F., Soler J., High-field limit of the Vlasov-Poisson-Fokker-Planck system, Arch. Ration. Mech. Anal. 158 (2001) 29-59. | MR 1834113 | Zbl 1038.82068

[36] O'Dwyer B., Victory H.D., On classical solutions of the Vlasov-Poisson-Fokker-Planck system, Indiana Univ. Math. J. 39 (1) (1990) 105-156. | MR 1052014 | Zbl 0674.60097

[37] Poupaud F., Runaway phenomena and fluid approximation under high fields in semiconductors kinetic theory, Z. Angew. Math. Mech. 72 (1992) 359-372. | MR 1178932 | Zbl 0785.76067

[38] Poupaud F., Soler J., Parabolic limit and stability of the Vlasov-Poisson-Fokker-Planck system, Math. Models Methods Appl. Sci. 10 (7) (2000) 1027-1045. | MR 1780148 | Zbl 1018.76048

[39] Puel M., Saint-Raymond L., Quasineutral limit for the relativistic Vlasov-Maxwell system, Asymptotic Anal. 40 (2004) 303-352. | MR 2107635 | Zbl 1072.35181

[40] Saint-Raymond L., Convergence of solutions to the Boltzmann equation in the incompressible Euler limit, Arch. Ration. Mech. Anal. 166 (2003) 47-80. | MR 1952079 | Zbl 1016.76071

[41] Vasseur A., Recent results on hydrodynamic limits, in: Dafermos C.M., Pokorny M. (Eds.), Handbook of Differential Equations: Evolutionary Equations, vol. 4, Elsevier, 2008. | MR 2508169 | Zbl pre05635159

[42] Victory H.D., On the existence of global weak solutions for the Vlasov-Poisson-Fokker-Planck system, J. Math. Anal. Appl. 160 (2) (1991) 525-555. | MR 1126136 | Zbl 0764.35024

[43] Wollman S., An existence and uniqueness theorem for the Vlasov-Maxwell system, Comm. Pure Appl. Math. 37 (1984) 457-462. | MR 745326 | Zbl 0592.45010

[44] Yau H.T., Relative entropy and hydrodynamics of Ginzburg-Landau models, Lett. Math. Phys. 22 (1) (1991) 63-80. | MR 1121850 | Zbl 0725.60120

[45] Yu H., Global classical solution of the Vlasov-Maxwell-Landau system near Maxwellians, J. Math. Phys. 45 (11) (2004) 4360-4376. | MR 2098143 | Zbl 1064.82035