×

Asymptotic and spectral analysis of the gyrokinetic-waterbag integro-differential operator in toroidal geometry. (English) Zbl 1348.35271

In this article the authors consider a model of plasma described by kinetic equations, amenable to rigorous analysis. It uses particular solutions of the Vlassov-Poisson (Vlassov-Maxwell) systems: the so called “waterbag” solutions. A spectral analysis of the problem is presented.

MSC:

35Q83 Vlasov equations
47G20 Integro-differential operators
82D10 Statistical mechanics of plasmas
82D75 Nuclear reactor theory; neutron transport
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
76U05 General theory of rotating fluids
78A35 Motion of charged particles
35Q60 PDEs in connection with optics and electromagnetic theory

Software:

GMWB3D-SLC
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions (1964) · Zbl 0171.38503
[2] Bardos, C.; Besse, N., The Cauchy problem for the Vlasov-Dirac-Benney equation and related issues in fluid mechanics and semi-classical limits, Kinet. Relat. Models, 6, 893-917 (2013) · Zbl 1292.35293 · doi:10.3934/krm.2013.6.893
[3] Berk, H. L.; Roberts, K. V., Nonlinear study of Vlasov’s equation for a special class of distribution functions, Phys. Fluids, 10, 1595-1597 (1967) · doi:10.1063/1.1762331
[4] Berk, H. L.; Nielsen, C. E.; Roberts, K. V., Phase space hydrodynamics of equivalent nonlinear systems: Experimental and computational observations, Phys. Fluids, 13, 980-995 (1970) · Zbl 0212.29802 · doi:10.1063/1.1693039
[5] Besse, N., On the Cauchy problem for the gyro-water-bag model, Math. Models Methods Appl. Sci., 21, 1839-1869 (2011) · Zbl 1230.35006 · doi:10.1142/S0218202511005623
[6] Besse, N., On the waterbag continuum, Arch. Ration. Mech. Anal., 199, 453-491 (2011) · Zbl 1229.35297 · doi:10.1007/s00205-010-0392-9
[7] Besse, N., Global weak solutions for the relativistic waterbag continuum, Math. Models Methods Appl. Sci., 21, 1150001 (2012) · Zbl 1246.35063 · doi:10.1142/S0218202512005848
[8] Besse, N.; Bertrand, P., Quasi-linear analysis of the gyro-water-bag model, Europhys. Lett., 83, 25003 (2008) · doi:10.1209/0295-5075/83/25003
[9] Besse, N.; Bertrand, P., Gyro-water-bag approach in nonlinear gyrokinetic turbulence, J. Comput. Phys., 228, 3973-3995 (2009) · Zbl 1273.82071 · doi:10.1016/j.jcp.2009.02.025
[10] Besse, N.; Berthelin, F.; Brenier, Y.; Bertrand, P., The multi-waterbag equations for collisionless kinetic modelling, Kinet. Relat. Models, 2, 39-90 (2009) · Zbl 1185.35292 · doi:10.3934/krm.2009.2.39
[11] Besse, N.; Mauser, N. J.; Sonnendrücker, E., Numerical approximation of self-consistent Vlasov models for low-frequency electromagnetic phenomena, Int. J. Appl. Math. Comput. Sci., 17, 101-114 (2007) · Zbl 1149.82028 · doi:10.2478/v10006-007-0030-3
[12] Brenier, Y., Une application de la symétrisation de Steiner aux equations hyperboliques: La méthode de transport et écroulement, C. R. Acad. Sci. Paris Ser. I Math., 292, 563-566 (1981) · Zbl 0459.35006
[13] Brenier, Y., Résolution d’équations d’évolution quasilinéaires en dimension N d’espace à l’aide d’équations linéaires en dimension N + 1, J. Differ. Equations, 50, 375-390 (1983) · Zbl 0549.35055 · doi:10.1016/0022-0396(83)90067-0
[14] Brenier, Y., Averaged multivalued solutions for scalar conservation laws, SIAM J. Numer. Anal., 21, 1013-1037 (1984) · Zbl 0565.65054 · doi:10.1137/0721063
[15] Brenier, Y.; Corrias, L., A kinetic formulation for multi-branch entropy solutions of scalar conservation laws, Ann. Inst. Henri Poincare Anal. Nonlinear, 15, 169-190 (1998) · Zbl 0893.35068 · doi:10.1016/S0294-1449(97)89298-0
[16] Brizard, A. J., New variational principle for the Vlasov-Maxwell equations, Phys. Rev. Lett., 84, 5768-5771 (2000) · doi:10.1103/PhysRevLett.84.5768
[17] Brizard, A. J.; Hahm, T. S., Foundations of nonlinear gyrokinetic theory, Rev. Mod. Phys., 79, 421-468 (2007) · Zbl 1205.76309 · doi:10.1103/RevModPhys.79.421
[18] Brown, B. M.; McCormack, D. K. R.; Evans, W. D.; Plum, M., On the spectrum of the second-order differential operators with complex coefficients, Proc. R. Soc. London A, 455, 1235-1257 (1999) · Zbl 0944.34018 · doi:10.1098/rspa.1999.0357
[19] Carleman, T., Zur theorie der linearen integralgleichungen, Math. Z., 9, 196-217 (1921) · JFM 48.1249.01 · doi:10.1007/BF01279029
[20] Case, K. M., Plasma oscillations, Ann. Phys., 7, 349-364 (1959) · Zbl 0096.44802 · doi:10.1016/0003-4916(59)90029-6
[21] Chance, M. S.; Dewar, R. L.; Frieman, E. A.; Glasser, A. H.; Greene, J. M.; Grimm, R. C.; Jardin, S. C.; Johnson, J. L.; Manickam, J.; Okabayashi, M.; Todd, A. M. M., MHD Stability limits on high-β tokamaks, 677-687 (1979)
[22] Connor, J. W.; Hastie, R. J.; Taylor, J. B., High mode number stability of an axisymmetric toroidal plasma, Proc. R. Soc. London A, 365, 1-17 (1979) · doi:10.1098/rspa.1979.0001
[23] Connor, J. W.; Taylor, J. B., Ballooning modes or Fourier modes in a toroidal plasma ?, Phys. Fluids, 30, 3180-3185 (1987) · Zbl 0636.76042 · doi:10.1063/1.866493
[24] Coulette, D. and Besse, N., “Numerical resolution of the global eigenvalue problem for gyrokinetic-waterbag model in toroidal geometry” (submitted). · Zbl 1348.35271
[25] Crandall, M. G.; Rabinowitz, P. H., Multiple solutions of a nonlinear integral equation, Arch. Ration. Mech. Anal., 37, 262-267 (1970) · Zbl 0194.14001 · doi:10.1007/BF00251607
[26] Davies, E. B., Linear Operators and Their Spectra, 106 (2007) · Zbl 1138.47001
[27] DePackh, D. C., The water-bag model of a sheet electron beam, J. Electron. Control, 13, 417-424 (1962) · doi:10.1080/00207216208937448
[28] Dewar, R. L.; Glasser, A. H., Ballooning mode spectrum in general toroidal systems, Phys. Fluids, 26, 3038-3052 (1983) · Zbl 0538.76123 · doi:10.1063/1.864028
[29] Dimits, A. M.; Bateman, G.; Beer, M. A.; Cohen, B. I.; Dorland, W.; Hammett, G. W.; Kim, C.; Kinsey, J. E.; Kotschenreuther, M.; Kritz, A. H.; Lao, L. L.; Mandrekas, J.; Nevins, W. M.; Parker, S. E.; Redd, A. J.; Shumaker, D. E.; Sydora, R.; Weiland, J., Comparisons and physics basis of tokamak transport models and turbulence simulations, Phys. Plasmas, 7, 969-983 (2003) · doi:10.1063/1.873896
[30] Dubin, D. H. E.; Krommes, J. A.; Oberman, C.; Lee, W. W., Nonlinear gyrokinetic equations, Phys. Fluids, 26, 3524-3535 (1983) · Zbl 0544.76158 · doi:10.1063/1.864113
[31] 31.Dunford, N. and Schwartz, J. T., Linear Operators, Part I: General Theory, Monographs on Pure and Applied Mathematics Vol. 7 (Interscience Publishers, 1958);Dunford, N. and Schwartz, J. T., Linear Operators, Part II: Spectral Theory, Monographs on Pure and Applied Mathematics Vol. 7 (Interscience Publishers, 1963). · Zbl 0084.10402
[32] Edmunds, D. E.; Evans, W. D., Spectral Theory and Differential Operators (1987) · Zbl 0628.47017
[33] Edwards, R. E., Functional Analysis, Theory and Applications (1965) · Zbl 0182.16101
[34] Eisenfeld, J., Operator equations and nonlinear eigenparameter problems, J. Funct. Anal., 12, 475-490 (1973) · Zbl 0255.47018 · doi:10.1016/0022-1236(73)90007-4
[35] Fedoryuk, M. V., Asymptotic Analysis (1993)
[36] Friedman, A.; Shinbrot, M., Nonlinear eigenvalue problems, Acta Math., 121, 77-125 (1968) · Zbl 0162.45704 · doi:10.1007/BF02391910
[37] Fredholm, I., Sur une classe d’équations fonctionnelles, Acta Math., 27, 365-390 (1903) · JFM 34.0422.02 · doi:10.1007/BF02421317
[38] Frieman, E. A.; Chen, L., Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria, Phys. Fluids, 25, 502-508 (1982) · Zbl 0506.76133 · doi:10.1063/1.863762
[39] Frieman, E. A.; Rewoldt, G.; Tang, W. M.; Glasser, A. H., General theory of kinetic ballooning modes, Phys. Fluids, 23, 1750-1769 (1980) · Zbl 0459.76095 · doi:10.1063/1.863201
[40] Garbet, X.; Idomura, Y.; Villard, L.; Watanabe, T. H., Gyrokinetic simulation of turbulent transport, Nucl. Fusion, 50, 043002 (2010) · doi:10.1088/0029-5515/50/4/043002
[41] Gel’fand, I. M.; Shilov, G. E., Generalized Functions: Properties and Operations, 1 (1964) · Zbl 0115.33101
[42] Gel’fand, I. M.; Shilov, G. E., Generalized Functions: Spaces of Fundamental and Generalized Functions, 2 (1968) · Zbl 0159.18301
[43] Giga, Y.; Miyakawa, T., A kinetic construction of global solutions of first order quasilinear equations, Duke Math. J., 50, 505-515 (1983) · Zbl 0519.35053 · doi:10.1215/S0012-7094-83-05022-6
[44] Glasser, A. H.; Coppi, B.; Sadowski, W., Ballooning modes in axisymmetric toroidal plasmas, 55-65 (1979)
[45] Glazman, I. M., Direct methods of qualitative spectral analysis of singular differential operators (1965) · Zbl 0143.36505
[46] Gohberg, I.; Goldberg, S.; Krupnik, N., Traces and determinants of linear operators, Integr. Equat. Oper. Theory, 26, 136-187 (1996) · Zbl 0859.47008 · doi:10.1007/BF01191855
[47] Gohberg, I.; Goldberg, S.; Krupnik, N., Hilbert-Carleman and regularized determinants for linear operators, Integr. Equat. Oper. Theory, 27, 10-47 (1997) · Zbl 0918.47020 · doi:10.1007/BF01195742
[48] Gohberg, I.; Goldberg, S.; Krupnik, N., Traces and Determinants for Linear Operators, Operator Theory: Advances and Applications, 116 (2000) · Zbl 0946.47013
[49] Gohberg, I.; Goldberg, S.; Kaashoek, M., Classes of Linear Operators, 1 (1990) · Zbl 0745.47002
[50] Gohberg, I.; Krein, M. G., Introduction to the Theory of Linear Nonselfadjoint Operators, 18 (1969) · Zbl 0181.13504
[51] Golub, G. H.; Van Der Vorst, H. A., Eigenvalue computation in the 20th century, J. Comput. Appl. Math., 123, 35-65 (2000) · Zbl 0965.65057 · doi:10.1016/S0377-0427(00)00413-1
[52] Goodwin, B. E., Integral equations with nonlinear eigenvalue parameters, SIAM Rev., 7, 368-394 (1965) · Zbl 0139.29302 · doi:10.1137/1007074
[53] Goodwin, B. E., Integral equations with nonlinear eigenvalue parameters, SIAM J. Appl. Math., 14, 65-85 (1966) · Zbl 0142.09203 · doi:10.1137/0114006
[54] Grandgirard, V.; Sarazin, Y.; Angelino, P.; Bottino, A.; Crouseilles, N.; Darmet, G.; Dif-Pradalier, G.; Garbet, X.; Ghendrih, P.; Jolliet, S.; Latu, G.; Sonnendrücker, E.; Villard, L., Global full-f gyrokinetic simulations of plasma turbulence, Plasma Phys. Controlled Fusion, 49, B173-B182 (2007) · doi:10.1088/0741-3335/49/12B/S16
[55] Grenier, E., Oscillations in quasineutral plasmas, Commun. Partial Differ. Equations, 21, 363-394 (1996) · Zbl 0849.35107 · doi:10.1080/03605309608821189
[56] Grenier, E., Limite quasineutre en dimension 1, Journées Equations Aux Dérivées Partielles (Saint-Jean-de-Monts, 1999) (1999) · Zbl 1008.35054
[57] Guillaume, P., Nonlinear eigenproblems, SIAM J. Matrix Anal. Appl., 20, 575-595 (1999) · Zbl 0930.65060 · doi:10.1137/S0895479897324172
[58] Hahm, T. S., Nonlinear gyrokinetic equations for tokamak microturbulence, Phys. Fluids, 31, 2670-2673 (1988) · Zbl 0649.76067 · doi:10.1063/1.866544
[59] Hazeltine, R. D.; Hitchcock, D. A.; Mahajan, S. M., Uniqueness and inversion of the ballooning representation, Phys. Fluids, 24, 180-181 (1981) · Zbl 0467.76117 · doi:10.1063/1.863238
[60] Hazeltine, R. D.; Meiss, J. D., Plasma Confinement (2003)
[61] Hazeltine, R. D.; Newcomb, W. A., Inversion of the ballooning transformation, Phys. Fluids B, 2, 7-10 (1990) · doi:10.1063/1.859490
[62] Hellinger, E.; Toeplitz, O., Integralgleichungen und gleichungen mit unendlichvielen unbekannien, Encyklopädie Math. Wiss., 2, 1335-1601 (1928) · JFM 54.0412.04
[63] Hilbert, D., Grundzüge einer allgemeinen theorie der linearen integralgleichungen (1912) · JFM 43.0423.01
[64] Hille, E.; Tamarkin, J. D., On the theory of linear integral equations I, Ann. Math., 31, 479-527 (1930) · JFM 56.0337.01 · doi:10.2307/1968241
[65] Hille, E.; Tamarkin, J. D., On the characteristic values of linear integral equations, Acta Math., 57, 1-76 (1931) · Zbl 0003.00902 · doi:10.1007/BF02403043
[66] Hille, E.; Tamarkin, J. D., On the theory of linear integral equations II, Ann. Math., 35, 445-455 (1934) · Zbl 0009.40201 · doi:10.2307/1968742
[67] Hohl, F.; Feix, M., Numerical experiments with a one-dimensional model for a self-gravitating star system, Astrophys. J., 147, 1164-1180 (1967) · doi:10.1086/149106
[68] Iglish, R., Über lineare integralgleichungen mit vom parameter abhängigem kern, Math. Ann., 117, 129-139 (1939) · JFM 65.0466.02
[69] Kato, T., Perturbation theory for linear operators, Grundlehren der Mathematischen Wissenschaften, 132 (1966) · Zbl 0148.12601
[70] Kim, J. Y.; Wakatani, M., Radial structure of high-mode-number toroidal modes in general equilibrium profiles, Phys. Rev. Lett., 73, 2200-2203 (1994) · doi:10.1103/PhysRevLett.73.2200
[71] Krasnosel’skii, M. A.; Pustylnik, E. I.; Sobolevskii, P. E.; Zabreiko, P. P., Integral Operators in Spaces of Summable Functions (1976) · Zbl 0312.47041
[72] Lee, Y. C.; Van Dam, J. W.; Coppi, B.; Sadowski, W., Kinetic theory of ballooning instabilities, 93-101 (1979)
[73] Lee, Y. C.; Van Dam, J. W.; Drake, J. F.; Lin, A. T.; Pritchett, P. L.; D’Ippolito, D.; Liewer, P. C.; Liu, C. S., Kinetic theory of ballooning instabilities and studies of tearing instabilities, 799-807 (1979)
[74] Lions, P.-L.; Perthame, B.; Tadmor, E., A kinetic formulation of multidimensional scalar conservation laws and related equations, J. Am. Math. Soc., 7, 169-191 (1994) · Zbl 0820.35094 · doi:10.1090/S0894-0347-1994-1201239-3
[75] Lions, P.-L.; Perthame, B.; Tadmor, E., Kinetic formulation of isentropic gas dynamics and p-systems, Commun. Math. Phys., 163, 415-431 (1994) · Zbl 0799.35151 · doi:10.1007/BF02102014
[76] Muskhelvlishvili, N. I., Singular Integral Equations (1965)
[77] Nayfeh, A. H., Perturbation Methods (2000) · Zbl 0995.35001
[78] Newcomb, W. A., Ballooning transformation, Phys. Fluids B, 2, 86-96 (1990) · doi:10.1063/1.859491
[79] Nevanlinna, R.; Paatero, V., Introduction to Complex Analysis (1969) · Zbl 0169.09001
[80] Paley, R. C.; Wiener, N., Fourier transforms in the complex plane, 19 (1934) · JFM 60.0345.02
[81] Perthame, B.; Tadmor, E., A kinetic equation with kinetic entropy functions for scalar conservation laws, Commun. Math. Phys., 136, 501-517 (1991) · Zbl 0729.76070 · doi:10.1007/BF02099071
[82] Perthame, B.; Tadmor, E., Kinetic Formulation of Conservation Laws (2003)
[83] Pogorzelski, W., Integral Equations and Their Applications (1966) · Zbl 0137.30502
[84] Plemelj, J., Zur theorie fredholmshen funktionalgleichung, Monatsh. Math. Phys., 15, 93-128 (1907) · JFM 35.0775.01 · doi:10.1007/BF01692293
[85] Qi, J.; Zheng, Z.; Sun, H., Classification of Sturm-Liouville differential equations with complex coefficients and operator realization, Proc. R. Soc. A, 467, 1835-1850 (2011) · Zbl 1228.34045 · doi:10.1098/rspa.2010.0281
[86] Rabinowitz, P. H., Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7, 487-513 (1971) · Zbl 0212.16504 · doi:10.1016/0022-1236(71)90030-9
[87] Rabinowitz, P. H., Some aspects of nonlinear eigenvalue problems, Rocky Mount. J. Math., 7, 161-202 (1973) · Zbl 0255.47069 · doi:10.1216/RMJ-1973-3-2-161
[88] Reed, M.; Simon, B., Methods of Modern Mathematical Physics IV: Analysis of Operators (1978) · Zbl 0401.47001
[89] Reed, M.; Simon, B., Methods of Modern Mathematical Physics I: Functional Analysis (1980) · Zbl 0459.46001
[90] Rewoldt, G.; Tang, W. M.; Frieman, E. A., Integral formulation for the two-dimensional spatial structure of drift and trapped-electron modes, Phys. Fluids, 21, 1513-1532 (1978) · Zbl 0385.76098 · doi:10.1063/1.862416
[91] Ruhe, A., Algorithms for the nonlinear eigenvalue problem, SIAM J. Numer. Anal., 10, 674-689 (1973) · Zbl 0261.65032 · doi:10.1137/0710059
[92] Sarazin, Y.; Grandgirard, V.; Fleurence, E.; Garbet, X.; Ghendrih, Ph.; Bertrand, P.; Depret, G., Kinetic features of interchange turbulence, Plasma Phys. Controlled Fusion, 47, 1817-1839 (2005) · doi:10.1088/0741-3335/47/10/013
[93] Sims, A. R., Secondary conditions for linear differential operators of the second order, J. Math. Mech., 6, 247-285 (1957) · Zbl 0077.29201
[94] Smithies, F., The Fredholm theory of integral equations, Duke Math. J., 8, 107-130 (1941) · JFM 67.0376.02 · doi:10.1215/S0012-7094-41-00805-0
[95] Smithies, F., Integral Equations (1958) · Zbl 0082.31901
[96] Steinberg, S., Meromorphic families of compact operators, Arch. Ration. Mech. Anal., 31, 372-379 (1968) · Zbl 0167.43002 · doi:10.1007/BF00251419
[97] Tamarkin, J. D., On Fredholm integral equations whose kernels are analytic in a parameter, Ann. Math., 28, 127-152 (1927) · JFM 53.0351.01 · doi:10.2307/1968363
[98] Tang, W. M., Microinstabilities theory in tokamaks, Nucl. Fusion, 18, 1089-1160 (1978) · doi:10.1088/0029-5515/18/8/006
[99] Tang, W. M.; Connor, J. W.; Hastie, R. J., Kinetic-ballooning mode theory in general geometry, Nucl. Fusion, 20, 1439-1453 (1980) · doi:10.1088/0029-5515/20/11/011
[101] Tisseur, F.; Meerbergen, K., The quadratic eigenvalue problem, SIAM Rev., 43, 235-286 (2001) · Zbl 0985.65028 · doi:10.1137/S0036144500381988
[102] Titchmarsh, E. C., Introduction to the Theory of Fourier Integrals (1948) · JFM 63.0367.05
[103] Turner, R. E. L., A class of nonlinear eigenvalue problems, J. Funct. Anal., 7, 297-322 (1968) · Zbl 0169.17004 · doi:10.1016/0022-1236(68)90009-8
[104] Turner, R. E. L., Nonlinear eigenvalue problems with nonlocal operators, Commun. Pure Appl. Math., 23, 963-972 (1970) · Zbl 0201.14501 · doi:10.1002/cpa.3160230607
[105] Van Kampen, N. G., On the theory of stationary waves in plasmas, Physica, 21, 949-963 (1955) · doi:10.1016/S0031-8914(55)93068-8
[106] Zakharov, V. E., Benney equations and quasiclassical approximation in the method of the inverse problem, Funct. Anal. Appl., 14, 89-98 (1980) · Zbl 0473.35075 · doi:10.1007/BF01086549
[107] Zettl, A., Sturm-Liouville Theory, 121 (2005) · Zbl 1103.34001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.