Anisotropie dans un plasma fortement magnétisé

Han-Kwan, Daniel

Séminaire Laurent Schwartz — EDP et applications (2011-2012), Exposé no. 25, 14 p.

Nous présentons les résultats prouvés dans [20, 22], qui concernent l’étude asymptotique de l’équation de Vlasov-Poisson dans un régime quasineutre et de champ magnétique intense. Nous insisterons en particulier sur les conséquences de l’anisotropie du problème physique sur l’analyse mathématique.

[1] G. Allaire. Homogenization and two-scale convergence. SIAM J. Math. Anal., 23(6) :1482–1518, 1992. | MR | Zbl

[2] M. Bostan. The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime. Asymptot. Anal., 61(2) :91–123, 2009. | MR | Zbl

[3] Y. Brenier. A homogenized model for vortex sheets. Arch. Rational Mech. Anal., 138(4) :319–353, 1997. | MR | Zbl

[4] Y. Brenier. Convergence of the Vlasov-Poisson system to the incompressible Euler equations. Comm. Partial Differential Equations, 25(3-4) :737–754, 2000. | MR | Zbl

[5] Y. Brenier and E. Grenier. Limite singulière du système de Vlasov-Poisson dans le régime de quasi neutralité : le cas indépendant du temps. C. R. Acad. Sci. Paris Sér. I Math., 318(2) :121–124, 1994. | MR | Zbl

[6] S. Cordier, E. Grenier, and Y. Guo. Two-stream instabilities in plasmas. Methods Appl. Anal., 7(2) :391–405, 2000. Cathleen Morawetz : a great mathematician. | MR | Zbl

[7] R. J. DiPerna and P.-L. Lions. Global weak solutions of Vlasov-Maxwell systems. Comm. Pure Appl. Math., 42(6) :729–757, 1989. | MR | Zbl

[8] R. J. DiPerna, P.-L. Lions, and Y. Meyer. $L^p$ regularity of velocity averages. Ann. Inst. H. Poincaré Anal. Non Linéaire, 8(3-4) :271–287, 1991. | Numdam | MR | Zbl

[9] E. Frénod and A. Mouton. Two-dimensional finite Larmor radius approximation in canonical gyrokinetic coordinates. J. Pure Appl. Math. Adv. Appl., 4(2) :135–169, 2010. | MR | Zbl

[10] E. Frénod and E. Sonnendrücker. The finite Larmor radius approximation. SIAM J. Math. Anal., 32(6) :1227–1247 (electronic), 2001. | MR | Zbl

[11] P. Ghendrih, M. Hauray, and A. Nouri. Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solutions. Kinet. and Relat. Models, 2(4) :707–725, 2009. | MR | Zbl

[12] F. Golse, P.L. Lions, B. Perthame, and R. Sentis. Regularity of the moments of the solution of a transport equation. J. Funct. Anal., 76(1) :110–125, 1988. | MR | Zbl

[13] F. Golse and L. Saint-Raymond. The Vlasov-Poisson system with strong magnetic field. J. Math. Pures Appl. (9), 78(8) :791–817, 1999. | MR | Zbl

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

[15] V. Grandgirard et al. Global full-$f$ gyrokinetic simulations of plasma turbulence. Plasma Phys. Control. Fusion, 49 :173–182, 2007.

[16] E. Grenier. Defect measures of the Vlasov-Poisson system in the quasineutral regime. Comm. Partial Differential Equations, 20(7-8) :1189–1215, 1995. | MR | Zbl

[17] E. Grenier. Oscillations in quasineutral plasmas. Comm. Partial Differential Equations, 21(3-4) :363–394, 1996. | MR | Zbl

[18] E. Grenier. Limite quasineutre en dimension 1. In Journées “Équations aux Dérivées Partielles” (Saint-Jean-de-Monts, 1999), pages Exp. No. II, 8. Univ. Nantes, Nantes, 1999. | MR | Zbl

[19] Y. Guo and W. A. Strauss. Nonlinear instability of double-humped equilibria. Ann. Inst. H. Poincaré Anal. Non Linéaire, 12(3) :339–352, 1995. | Numdam | MR | Zbl

[20] D. Han-Kwan. The three-dimensional finite Larmor radius approximation. Asymptot. Anal., 66(1) :9–33, 2010. | MR | Zbl

[21] D. Han-Kwan. Effect of the polarization drift in a strongly magnetized plasma. ESAIM Math. Mod. Num. Anal., 46(4) :929-947, 2012. | Numdam | MR

[22] D. Han-Kwan. On the three-dimensional finite Larmor radius approximation : the case of electrons in a fixed background of ions. Soumis, 2011. | MR | Zbl

[23] D. Han-Kwan. Quasineutral limit of the Vlasov-Poisson system with massless electrons. Comm. Partial Differential Equations, 36(8) :1385–1425, 2011. | MR | Zbl

[24] M. Hauray and A. Nouri. Well-posedness of a diffusive gyro-kinetic model. To appear in Ann. IHP (Analyse Non Linéaire), 2011. | Numdam | MR

[25] P.-E. Jabin. Averaging lemmas and dispersion estimates for kinetic equations. Riv. Mat. Univ. Parma (8), 1 :71–138, 2009. | MR | Zbl

[26] G. Loeper. Quasi-neutral limit of the Euler-Poisson and Euler-Monge-Ampère systems. Comm. Partial Differential Equations, 30(7-9) :1141–1167, 2005. | MR | Zbl

[27] N. Masmoudi. From Vlasov-Poisson system to the incompressible Euler system. Comm. Partial Differential Equations, 26(9) :1913–1928, 2001. | MR | Zbl

[28] C. Mouhot and C. Villani. On Landau damping. To appear in Acta Mathematica, 2011. | MR | Zbl

[29] G. Nguetseng. A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal., 20(3) :608–623, 1989. | MR | Zbl

[30] L. Nirenberg. An abstract form of the nonlinear Cauchy-Kowalewski theorem. J. Differential Geom., 6 :561–576, 1972. | MR | Zbl

[31] T. Nishida. A note on a theorem of Nirenbeg. J. Differential Geom., 12 :629–633, 1977. | MR | Zbl

[32] O. Penrose. Electrostatic instability of a uniform non-Maxwellian plasma. Phys. Fluids, 3 :258–265, 1960. | Zbl