The Vlasov-Poisson system with strong magnetic field in quasineutral regime

Mathematical Models and Methods in Applied Sciences

Volumn 13, Issue 5, 2003, Pages 661-714

  Golse, F ^a   Saint Raymond, L ^b  

^a ECOLE NORMALE SUPÉRIEURE   (France)

^b SORBONNE UNIVERSITÉ   (France)

Author keywords

Gyrokinetic approximation; High frequency filtering; Quasineutral limit; Relative entropy method; Vlasov Poisson

Indexed keywords

EID: 0038407473     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202503002647     Document Type: Article

References (29)

1. A. A. Arsen'ev, Existence in the large of a weak solution of Vlasov's system of equations, Z. Vyčisl. Mat. Mat. Fiz. 15 (1975) 136-147.
2. K. Asano and S. Ukai, On the Vlasov-Poisson limit of the Vlasov-Maxwell equation. Pattern and waves, Qualitative analysis of nonlinear differential equations, Stud. Math. Appl. 18 (1986) 369-383.
3. A. Babin, A. Mahalov and B. Nicolaenko, Global splitting, integrability and regularity of 3D Euler and Navier-Stokes equations for uniformly rotating fluids, Euro. J. Mech. 15 (1996) 291-300.
4. A. Babin, A. Mahalov and B. Nicolaenko, Regularity and integrability of 3D Euler and Navier-Stokes equations for rotating fluids, Asympt. Anal. 15 (1997) 103-150.
5. F. Bouchut, F. Golse and M. Pulvirenti, Kinetic Equations and Asymptotic Theory, Series in Applied Math. No. 4 (Gauthier-Villars, 2000).
6. Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. P.D.E. 25 (2000) 737-754.
7. Y. Brenier, A remark on the low Rossby number theory, Internal Report of the Newton Institute (1996).
8. J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Fluids with anisotropic viscosity, Special issue for R. Temam's 60th birthday, Math. Model. Numer. Anal. 34 (2000) 315-335.
9. P. Degond, Local existence of solutions of the Vlasov-Maxwell equations and convergence to the Vlasov-Poisson equations for infinite light velocity, Math. Methods Appl. Sci. 8 (1986) 533-558.
10. J.-M. Delort, Existence de nappes de tourbillon en dimension deux, J. Amer. Math. Soc. 4 (1991), 553-586.
11. B. Desjardins and E. Grenier, Derivation of quasi-geostrophic potential vorticity equations, Adv. Differential Equations 3 (1998) 715-752.
12. R. DiPerna, Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys. 91 (1983) 1-30.
13. R. DiPerna and P.-L. Lions, Equations différentielles ordinaires et équations de transport avec des coefficients irréguliers, Séminaire sur les Equations aux Dérivées Partielles, 1988-1989, Exp. No. XIV, Ecole Polytech., Palaiseau, 1989.
14. E. Frénod and E. Sonnendrücker, Homogenization of the Vlasov equation and the Vlasov-Poisson system with a strong external magnetic field, Asympt. Anal. 18 (1998) 193-213.
15. E. Frénod and E. Sonnendrücker, Long time behavior of the two-dimensional Vlasov equation with a strong external magnetic field, Math. Models Methods Appl. Sci. 10 (2000) 539-553.
16. I. Gallagher, Applications of Schochet's methods to parabolic equations, J. Math. Pures Appl. 77 (1998) 989-1054.
17. I. Gallagher, Asymptotic of the solutions of hyperbolic equations with a skew-symmetric perturbation, J. Differential Equations 150 (1998) 363-384.
18. F. Golse and L. Saint-Raymond, The guiding center approximation for the Vlasov-Poisson System, J. Math. Pures Appl. 78 (1999) 791-817.
19. H. Grad, The guiding center plasma, Proc. Symp. Appl. Math. XVIII (1967) 162-248.
20. E. Grenier, Pseudo-differential energy estimates of singular perturbations, Comm. Pure Appl. Math. 50 (1997) 821-865.
21. E. Grenier, Oscillations in quasineutral plasmas, Comm. P.D.E. 21 (1996) 363-394.
22. E. Grenier, On the stability of boundary layers of incompressible Euler equations, J. Differential Equations 164 (2000) 180-222.
23. E. M. Lifshitz and L. P. Pitaevski, Course of Theoretical Physics, Vol. 10 (Pergamon Press, N.Y. 1981).
24. P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1: Incompressible Models, Oxford Lecture Series in Mathematics and its Applications (Clarendon Press, 1996).
25. P.-L. Lions and N. Masmoudi, From Boltzmann equations to the Stokes and Euler equations II, Arch. Rat. Mech. Anal. 158 (2001) 195-211.
26. L. Saint-Raymond, The gyrokinetic approximation for the Vlasov-Poisson system, Math. Models Methods Appl. Sci. 10 (2000) 1305-1332.
27. S. Schochet, Fast singular limits of hyperbolic PDEs, J. Differential Equations 114 (1994) 476-512.
28. E. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton Univ. Press, 1970).
29. S. Ukai and T. Okabe, On classical solutions in the large in time of two-dimensional Vlasov's equation, Osaka J. Math. 15 (1978) 245-261.