Well-posedness of a diffusive gyro-kinetic model
Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 4, p. 529-550

We study a finite Larmor radius model used to describe the ions distribution function in the core of a tokamak plasma, that consists in a gyro-kinetic transport equation coupled with an electro-neutrality equation. Since the last equation does not provide enough regularity on the electric potential, we introduce a simple linear collision operator adapted to the finite Larmor radius approximation. We next study the two-dimensional dynamics in the direction perpendicular to the magnetic field. Thanks to the smoothing effects of the collision and the gyro-average operators, we prove the global existence of solutions, as well as short time uniqueness and stability.

On étudie un modèle à rayon de Larmor fini décrivant la fonction de distribution des ions dans un plasma de coeur de tokamak. Il consiste en une équation de transport gyrocinétique couplée à une équation de quasi-neutralité. Lʼéquation de quasi-neutralité donnant peu de régularité au potentiel électrique, on introduit un opérateur de collisions linéaire adapté. On étudie alors la dynamique du système dans la direction perpendiculaire au champ magnétique. Lʼeffet régularisant des opérateurs de collisions et de gyro-moyenne permet de démontrer lʼexistence globale de solutions ainsi que leur unicité et stabilité locales en temps.

DOI : https://doi.org/10.1016/j.anihpc.2011.03.002
Classification:  41A60,  76P05,  82A70,  78A35
Keywords: Plasmas, Gyro-kinetic model, Electro-neutrality equation, Cauchy problem
     author = {Hauray, M. and Nouri, A.},
     title = {Well-posedness of a diffusive gyro-kinetic model},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {28},
     number = {4},
     year = {2011},
     pages = {529-550},
     doi = {10.1016/j.anihpc.2011.03.002},
     zbl = {1269.82075},
     mrnumber = {2823883},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2011__28_4_529_0}
Hauray, M.; Nouri, A. Well-posedness of a diffusive gyro-kinetic model. Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 4, pp. 529-550. doi : 10.1016/j.anihpc.2011.03.002. http://www.numdam.org/item/AIHPC_2011__28_4_529_0/

[1] F. Bouchut, Smoothing effect for the non-linear Vlasov–Poisson–Fokker–Planck system, J. Differential Equations 122 no. 2 (1995), 225-238 | MR 1355890 | Zbl 0840.35053

[2] A.J. Brizard, A guiding-center Fokker–Planck collision operator for non-uniform magnetic fields, Phys. Plasmas 11 (September 2004), 4429-4438 | MR 2095562

[3] S. Cordier, E. Grenier, Quasineutral limit of an Euler–Poisson system arising from plasma physics, Commun. Partial Differ. Equations 25 no. 5–6 (2000), 1099-1113 | MR 1759803 | Zbl 0978.82086

[4] S. Cordier, Y.-J. Peng, Système Euler–Poisson non linéaire. Existence globale de solutions faibles entropiques, RAIRO Modél. Math. Anal. Numér. 32 no. 1 (1998), 1-23 | Numdam | MR 1619591

[5] E. Frénod, É. Sonnendrücker, The finite Larmor radius approximation, SIAM J. Math. Anal. 32 no. 6 (2001), 1227-1247 | MR 1856246 | Zbl 0980.82030

[6] P. Ghendrih, M. Hauray, A. Nouri, Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solutions, Kinet. Relat. Models 2 no. 4 (2009), 707-725 | MR 2556718 | Zbl 1195.82087

[7] V. Grandgirard, Y. Sarazin, X. Garbet, G. Dif-Pradalier, P. Ghendrih, N. Crouseilles, G. Latu, E. Sonnendrücker, N. Besse, P. Bertrand, GYSELA, a full-f global gyrokinetic semi-Lagrangian code for ITG turbulence simulations, in: O. Sauter (Ed.), Theory of Fusion Plasmas, in: American Institute of Physics Conference Series, vol. 871, November 2006, pp. 100–111.

[8] O.A. Ladyženskaja, V.A. Solonnikov, N.N. UralʼCeva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs vol. 23, American Mathematical Society, Providence, RI (1967) | MR 241822

[9] L.D. Landau, The transport equation in the case of Coulomb interactions, D. Ter Haar (ed.), Collected Papers of L.D. Landau, Pergamon Press, Oxford (1981), 163-170

[10] J.-L. Lions, E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques vol. 17, Dunod, Paris (1968) | MR 291887 | Zbl 0165.10801

[11] J. Moser, A sharp form of an inequality by Trudinger, Indiana Univ. Math. J. 20 (1971), 1077-1092 | Zbl 0203.43701

[12] G.N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge (1995) | Zbl 0849.33001