Previous |  Up |  Next


Title: On a nonlocal problem for a confined plasma in a Tokamak (English)
Author: Zou, Weilin
Author: Li, Fengquan
Author: Lv, Boqiang
Language: English
Journal: Applications of Mathematics
ISSN: 0862-7940
Volume: 58
Issue: 6
Year: 2013
Pages: 609-642
Summary lang: English
Category: math
Summary: The paper deals with a nonlocal problem related to the equilibrium of a confined plasma in a Tokamak machine. This problem involves terms $u'_{\ast }(|u>u(x)|)$ and $|u>u(x)|$, which are neither local, nor continuous, nor monotone. By using the Galerkin approximate method and establishing some properties of the decreasing rearrangement, we prove the existence of solutions to such problem. (English)
Keyword: nonlinear elliptic equation; relative rearrangement; Tokamak; decreasing rearrangement; plasma physics
MSC: 35D99
MSC: 35M10
MSC: 35Q35
idZBL: Zbl 06312918
idMR: MR3162751
DOI: 10.1007/s10492-013-0031-5
Date available: 2013-11-09T20:15:30Z
Last updated: 2016-01-04
Stable URL:
Reference: [1] Jr., F. J. Almgren, Lieb, E. H.: Symmetric decreasing rearrangement is sometimes continuous.J. Am. Math. Soc. 2 (1989), 683-773. Zbl 0688.46014, MR 1002633, 10.1090/S0894-0347-1989-1002633-4
Reference: [2] Berestycki, H., Brézis, H.: On a free boundary problem arising in plasma physics.Nonlinear Anal., Theory Methods Appl. 4 (1980), 415-436. Zbl 0437.35032, MR 0574364, 10.1016/0362-546X(80)90083-8
Reference: [3] Blum, J.: Numerical Simulation and Optimal Control in Plasma Physics. With Applications to Tokamaks. Wiley/Gauthier-Villars Series in Modern Applied Mathematics.Wiley Chichester (1989). MR 0996236
Reference: [4] Boccardo, L., León, S. Segura de, Trombetti, C.: Bounded and unbounded solutions for a class of quasi-linear elliptic problems with a quadratic gradient term.J. Math. Pures Appl. 80 (2001), 919-940. MR 1865381, 10.1016/S0021-7824(01)01211-9
Reference: [5] Courant, R., Hilbert, D.: Methods of Mathematical Physics Vol. I. Translated and revised from the German original. First English ed.Interscience Publishers New York (1953). Zbl 0051.28802, MR 0065391
Reference: [6] Díaz, J. I., Galiano, G., Padial, J. F.: On the uniqueness of solutions of a nonlinear elliptic problem arising in the confinement of a plasma in a stellarator device.Appl. Math. Optimization 39 (1999), 61-73. Zbl 0923.35056, MR 1654558, 10.1007/s002459900098
Reference: [7] Díaz, J. I., Lerena, M. B., Padial, J. F., Rakotoson, J. M.: An elliptic-parabolic equation with a nonlocal term for the transient regime of a plasma in a stellarator.J. Differ. Equations 198 (2004), 321-355. Zbl 1050.35151, MR 2038584, 10.1016/j.jde.2003.07.015
Reference: [8] Díaz, J. I., Padial, J. F., Rakotoson, J. M.: Mathematical treatment of the magnetic confinement in a current carrying stellarator.Nonlinear Anal., Theory Methods Appl. 34 (1998), 857-887. Zbl 0946.35119, MR 1636600
Reference: [9] Díaz, J. I., Rakotoson, J. M.: On a nonlocal stationary free-boundary problem arising in the confinement of a plasma in a stellarator geometry.Arch. Ration. Mech. Anal. 134 (1996), 53-95. Zbl 0863.76092, MR 1392309, 10.1007/BF00376255
Reference: [10] Ferone, A., Jalal, M., Rakotoson, J. M., Volpicelli, R.: A topological approach for generalized nonlocal models for a confined plasma in a tokamak.Commun. Appl. Anal. 5 (2001), 159-181. Zbl 1084.35512, MR 1844189
Reference: [11] Ferone, A., Jalal, M., Rakotoson, J. M., Volpicelli, R.: Nonlocal generalized models for a confined plasma in a tokamak.Appl. Math. Lett. 12 (1999), 43-46. Zbl 0951.76099, MR 1663417, 10.1016/S0893-9659(98)00124-4
Reference: [12] Fiorenza, A., Rakotoson, J. M., Zitouni, L.: Relative rearrangement method for estimating dual norms.Indiana Univ. Math. J. 58 (2009), 1127-1150. Zbl 1178.46026, MR 2541361, 10.1512/iumj.2009.58.3580
Reference: [13] Gourgeon, H., Mossino, J.: Sur un problème à frontière libre de la physique des plasmas.Ann. Inst. Fourier 29 (1979), 127-141 French. Zbl 0405.35070, MR 0558592, 10.5802/aif.770
Reference: [14] Grad, H., Hu, P. N., Stevens, D. C.: Adiabatic evolution of plasma equilibrium.Proc. Nat. Acad. Sci. USA 72 (1975), 3789-3793. 10.1073/pnas.72.10.3789
Reference: [15] Henrot, A.: Extremum Problems for Eigenvalues of Elliptic Operators. Frontiers in Mathematics.Birkhäuser Basel (2006). MR 2251558
Reference: [16] Lieb, E. H., Loss, M.: Analysis. 2nd ed. Graduate Studies in Mathematics 14.American Mathematical Society Providence (2001). MR 1817225
Reference: [17] Mercier, C.: The Magnetohydrodynamic Approach to the Problem of a Plasma Confinement in Closed Magnetic Configurations.EURATOM-CEA, Comm. of the European Communities Luxembourg (1974).
Reference: [18] Mossino, J.: A priori estimates for a model of Grad-Mercier type in plasma confinement.Appl. Anal. 13 (1982), 185-207. Zbl 0478.35018, MR 0663773, 10.1080/00036818208839390
Reference: [19] Mossino, J.: Application des inéquations quasi-variationnelles à quelques problèmes non linéaires de la physique des plasmas.Isr. J. Math. 30 (1978), 14-50 French. Zbl 0399.35047, MR 0508250, 10.1007/BF02760826
Reference: [20] Mossino, J.: Inégalités isopérimétriques et applications en physique. Travaux en Cours.Hermann Paris (1984), French. MR 0733257
Reference: [21] Mossino, J.: Some nonlinear problems involving a free boundary in plasma physics.J. Differ. Equations 34 (1979), 114-138. Zbl 0406.35027, MR 0549587, 10.1016/0022-0396(79)90021-4
Reference: [22] Mossino, J., Temam, R.: Directional derivative of the increasing rearrangement mapping and application to a queer differential equation in plasma physics.Duke Math. J. 48 (1981), 475-495. Zbl 0476.35031, MR 0630581, 10.1215/S0012-7094-81-04827-4
Reference: [23] Mossino, J., Temam, R.: Free boundary problems in plasma physics: review of results and new developments.Free Boundary Problems, Theory and Applications Vol. II. Proc. interdisc. Symp., Montecatini/Italy 1981, Res. Notes Math. 79 A. Fasano Pitman (1983), 672-681. Zbl 0512.76126
Reference: [24] Rakotoson, J. M.: Existence of bounded solutions of some degenerate quasilinear elliptic equations.Commun. Partial Differ. Equations 12 (1987), 633-676. Zbl 0627.47035, MR 0879354, 10.1080/03605308708820505
Reference: [25] Rakotoson, J. M.: Galerkin approximation, strong continuity of the relative rearrangement map and application to plasma physics equations.Differ. Integral Equ. 12 (1999), 67-81. Zbl 1005.76097, MR 1668537
Reference: [26] Rakotoson, J. M.: Multivalued fixed point index and nonlocal problems involving relative rearrangement.Nonlinear Anal., Theory Methods Appl. 66 (2007), 2470-2499. Zbl 1130.46014, MR 2312600, 10.1016/
Reference: [27] Rakotoson, J. M.: Relative Rearrangement. An Estimation Tool for Boundary Problems. (Réarrangement relatif. Un instrument d'estimations dans les problèmes aux limites).Mathématiques & Applications 64 Springer, Berlin (2008), French. Zbl 1170.35036, MR 2455723, 10.1007/978-3-540-69118-1
Reference: [28] Rakotoson, J. M.: Relative rearrangement for highly nonlinear equations.Nonlinear Anal., Theory Methods Appl. 24 (1995), 493-507. Zbl 0830.35036, MR 1315691, 10.1016/0362-546X(95)93090-Q
Reference: [29] Rakotoson, J. M.: Un modèle non local en physique des plasmas: résolution par une méthode de degré topologique. (A nonlocal model in plasma physics: solution by the method of topological degree).Acta Appl. Math. 4 (1985), 1-14 French. Zbl 0586.35091, MR 0791260, 10.1007/BF02293489
Reference: [30] Rakotoson, J. M., Seoane, M. L.: Numerical approximations of the relative rearrangement: the piecewise linear case. Application to some nonlocal problems.M2AN, Math. Model. Numer. Anal. 34 (2000), 477-499. Zbl 0963.76052, MR 1765671, 10.1051/m2an:2000152
Reference: [31] Rakotoson, J. M., Temam, R.: A co-area formula with applications to monotone rearrangement and to regularity.Arch. Ration. Mech. Anal. 109 (1990), 213-238. Zbl 0735.49039, MR 1025171, 10.1007/BF00375089
Reference: [32] Stampacchia, G.: Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus.Ann. Inst. Fourier 15 (1965), 189-257 French. Zbl 0151.15401, MR 0192177, 10.5802/aif.204
Reference: [33] Temam, R.: A non-linear eigenvalue problem: the shape at equilibrium of a confined plasma.Arch. Ration. Mech. Anal. 60 (1975), 51-73. Zbl 0328.35069, MR 0412637, 10.1007/BF00281469
Reference: [34] Temam, R.: Monotone rearrangement of a function and the Grad-Mercier equation of plasma physics. Recent methods in non-linear analysis, Proc. Int. Meet., Rome 1978.Pitagora Bologna 83-98 (1979). MR 0533163
Reference: [35] Temam, R.: Remarks on a free boundary value problem arising in plasma physics.Commun. Partial Differ. Equations 2 (1977), 563-585. Zbl 0355.35023, MR 0602544, 10.1080/03605307708820039
Reference: [36] Trombetti, C.: Non-uniformly elliptic equations with natural growth in the gradient.Potential Anal. 18 (2003), 391-404. Zbl 1040.35010, MR 1953268, 10.1023/A:1021884903872
Reference: [37] Zou, W., Li, F., Lv, B.: On a nonlocal elliptic problem arising in the confinement of a plasma in a current carrying stellarator.Mathematical Methods in the Applied Sciences 36 (2013), 2128-2144. Zbl 1276.35146


Files Size Format View
AplMat_58-2013-6_1.pdf 362.8Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo