On the three-dimensional finite Larmor radius approximation: The case of electrons in a fixed background of ions

Han-Kwan, Daniel

Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013), p. 1127-1157 / Harvested from Numdam

Résumé

This paper is concerned with the analysis of a mathematical model arising in plasma physics, more specifically in fusion research. It directly follows, Han-Kwan (2010) [18], where the three-dimensional analysis of a Vlasov–Poisson equation with finite Larmor radius scaling was led, corresponding to the case of ions with massless electrons whose density follows a linearized Maxwell–Boltzmann law. We now consider the case of electrons in a background of fixed ions, which was only sketched in Han-Kwan (2010) [18]. Unfortunately, there is evidence that the formal limit is false in general. Nevertheless, we formally derive from the Vlasov–Poisson equation a fluid system for particular monokinetic data. We prove the local in time existence of analytic solutions and rigorously study the limit (when the inverse of the intensity of the magnetic field and the Debye length vanish) to a new anisotropic fluid system. This is achieved thanks to Cauchy–Kovalevskaya type techniques, as introduced by Caflisch (1990) [7] and Grenier (1996) [14]. We finally show that this approach fails in Sobolev regularity, due to multi-fluid instabilities.

Détail

Publié le : 2013-01-01

MR 3132419 | Zbl 1338.82055

DOI : https://doi.org/10.1016/j.anihpc.2012.12.012

Bibliographie

[1] A.A. Arsenev, Existence in the large of a weak solution of Vlasovʼs system of equations, Z. Vychisl. Mat. Mat. Fiz. 15 (1975), 136-147 | MR 371322

[2] N. Besse, F. Berthelin, Y. Brenier, P. Bertrand, The multi-water-bag equations for collisionless kinetic modeling, Kinet. Relat. Models 2 no. 1 (2009), 39-80 | MR 2472149 | Zbl 1185.35292

[3] M. Bostan, The Vlasov–Poisson system with strong external magnetic field. Finite Larmor radius regime, Asymptot. Anal. 61 no. 2 (2009), 91-123 | MR 2499194 | Zbl 1180.35501

[4] Y. Brenier, Some conservation laws given by kinetic models, J. EDP (1995), 1-13 | Numdam | Zbl 0873.35065

[5] Y. Brenier, A homogenized model for vortex sheets, Arch. Ration. Mech. Anal. 138 (1997), 319-353 | MR 1467558 | Zbl 0962.35140

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

[7] R. Caflisch, A simplified version of the abstract Cauchy–Kowalewski theorem with weak singularities, Bull. Amer. Math. Soc. (N.S.) 23 no. 2 (1990), 495-500 | MR 1027897 | Zbl 0723.35003

[8] S. Cordier, E. Grenier, Y. Guo, Two-stream instabilities in plasmas, Methods Appl. Anal. 7 no. 2 (2000), 391-405 | MR 1869291 | Zbl 1002.82029

[9] E. Frénod, A. Mouton, Two-dimensional finite Larmor radius approximation in canonical gyrokinetic coordinates, J. Pure Appl. Math.: Adv. Appl. 4 no. 2 (2010), 135-166 | MR 2816864 | Zbl 1225.35016

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

[11] 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

[12] F. Golse, L. Saint-Raymond, The Vlasov–Poisson system with strong magnetic field, J. Math. Pures. Appl. 78 (1999), 791-817 | MR 1715342 | Zbl 0977.35108

[13] V. Grandgirard, et al., Global full-f gyrokinetic simulations of plasma turbulence, Plasma Phys. Control. Fusion 49 (2007), 173-182

[14] E. Grenier, Oscillations in quasineutral plasmas, Comm. Partial Differential Equations 21 no. 3–4 (1996), 363-394 | MR 1387452 | Zbl 0849.35107

[15] E. Grenier, Limite quasineutre en dimension 1, Journées “Équations aux Dérivées Partielles”, Saint-Jean-de-Monts, 1999, Univ. Nantes, Nantes (1999) | MR 1718954

[16] Y. Guo, W.A. Strauss, Nonlinear instability of double-humped equilibria, Ann. I. H. P., Sect. C 12 no. 3 (1995), 339-352 | Numdam | MR 1340268 | Zbl 0836.35130

[17] D. Han-Kwan, On the confinement of a tokamak plasma, SIAM J. Math. Anal. 42 no. 6 (2010), 2337-2367 | MR 2733252 | Zbl 1229.82146

[18] D. Han-Kwan, The three-dimensional finite Larmor radius approximation, Asymptot. Anal. 66 no. 1 (2010), 9-33 | MR 2582446 | Zbl 1191.35267

[19] D. Han-Kwan, Contribution à lʼétude mathématique des plasmas fortement magnétisés, PhD thesis, 2011.

[20] D. Han-Kwan, Effect of the polarization drift in a strongly magnetized plasma, ESAIM Math. Model. Numer. Anal. 46 no. 4 (2012), 929-947 | Numdam | MR 2891475 | Zbl 1310.76190

[21] M. Hauray, A. Nouri, Well-posedness of a diffusive gyro-kinetic model, Ann. Inst. H. Poincaré Anal. Non Linéaire 28 no. 4 (2011), 529-550 | Numdam | MR 2823883 | Zbl 1269.82075

[22] T. Kato, On classical solutions of the two-dimensional nonstationary Euler equation, Arch. Ration. Mech. Anal. 25 (1967), 188-200 | MR 211057 | Zbl 0166.45302

[23] Z. Lin, S. Ethier, T.S. Hahm, W.M. Tang, Size scaling of turbulent transport in magnetically confined plasmas, Phys. Rev. Lett. 88 no. 19 (May 2002), 195004-1-195004-4

[24] Andrew J. Majda, Andrea L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts Appl. Math. vol. 27, Cambridge University Press, Cambridge (2002) | MR 1867882 | Zbl 0983.76001

[25] C. Mouhot, C. Villani, On Landau damping, Acta Math. 207 no. 1 (2011), 29-201 | MR 2863910 | Zbl 1239.82017

[26] O. Penrose, Electrostatic instability of a uniform non-Maxwellian plasma, Phys. Fluids 3 (1960), 258-265 | Zbl 0090.22801

[27] J. Simon, Compact sets in $ {L}^{p}(0,T;B)$, Ann. Mat. Pura. Appl. 146 (1987), 65-96 | MR 916688 | Zbl 0629.46031

[28] P.L. Sulem, Introduction to the guiding center theory. Topics in kinetic theory, Fields Inst. Commun., Amer. Math. Soc. 46 (2005), 109-149 | MR 2191923 | Zbl 1330.76150

[29] J. Wesson, Tokamaks, Clarendon Press, Oxford (2004) | Zbl 1111.82054

[30] V.I. Yudovič, Non-stationary flows of an ideal incompressible fluid, Z. Vychisl. Mat. Mat. Fiz. 3 (1963), 1032-1066 | MR 158189