Geodesic
Limit model for the Vlasov--Maxwell system with strong magnetic fields via gyroaveraging
Algebra i analiz, Tome 32 (2020) no. 4, pp. 200-216.

Voir la notice de l'article provenant de la source Math-Net.Ru

This paper deals with the Vlasov–Maxwell system in the case of a strong magnetic field. After a physically motivated nondimensionalization of the original system, a Hilbert expansion is employed around a small parameter given as the product of the characteristic time scale and the gyrofrequency. From this, necessary conditions on the solvability of the reduced system are derived. An important aspect is the reduction of the six-dimensional phase space to five dimensions. In addition to the discussion of the partial differential equations, also initial and boundary conditions both for the full system and the limit model are studied.
Keywords: Vlasov–Maxwell system, strong magnetic field, gyrokinetics. 
@article{AA_2020_32_4_a4,
     author = {T. Kessler and S. Rjasanow},
     title = {Limit model for the {Vlasov--Maxwell} system with strong magnetic fields via gyroaveraging},
     journal = {Algebra i analiz},
     pages = {200--216},
     publisher = {mathdoc},
     volume = {32},
     number = {4},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AA_2020_32_4_a4/}
}
TY  - JOUR
AU  - T. Kessler
AU  - S. Rjasanow
TI  - Limit model for the Vlasov--Maxwell system with strong magnetic fields via gyroaveraging
JO  - Algebra i analiz
PY  - 2020
SP  - 200
EP  - 216
VL  - 32
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AA_2020_32_4_a4/
LA  - en
ID  - AA_2020_32_4_a4
ER  - 
%0 Journal Article
%A T. Kessler
%A S. Rjasanow
%T Limit model for the Vlasov--Maxwell system with strong magnetic fields via gyroaveraging
%J Algebra i analiz
%D 2020
%P 200-216
%V 32
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AA_2020_32_4_a4/
%G en
%F AA_2020_32_4_a4
T. Kessler; S. Rjasanow. Limit model for the Vlasov--Maxwell system with strong magnetic fields via gyroaveraging. Algebra i analiz, Tome 32 (2020) no. 4, pp. 200-216. http://geodesic.mathdoc.fr/item/AA_2020_32_4_a4/

[1] Belyaeva Yu. O., “Statsionarnye resheniya uravnenii Vlasova dlya vysokotemperaturnoi dvukomponentnoi plazmy”, Sovremen. Mat. Fundam. Napravl., 62, 2016, 19–31

[2] Belyaeva Yu. O., Skubachevskii A. L., “Ob odnoznachnoi razreshimosti pervoi smeshannoi zadachi dlya sistemy uravnenii Vlasova-Puassona v beskonechnom tsilindre”, Zap. nauch. semin. POMI, 477, 2018, 12–34

[3] Boozer A. H., “Physics of magnetically confined plasmas”, Rev. Mod. Phys., 76 (2005), 1071–1141 | DOI

[4] Bostan M., “Gyrokinetic Vlasov equation in three dimensional setting. Second order approximation”, Multiscale Model. Simul., 8:5 (2010), 1923–1957 | DOI | MR | Zbl

[5] Brizard A. J., Hahm T. S., “Foundations of nonlinear gyrokinetic theory”, Rev. Mod. Phys., 79:2 (2007), 421–468 | DOI | MR | Zbl

[6] Cercignani C., The Boltzmann equation and its applications, Appl. Math. Sci., 67, Springer, New York, 1988 | DOI | MR | Zbl

[7] Chen F. F., Introduction to plasma physics and controlled fusion, Springer Int. Publ., 2016

[8] Degond P., Filbet F., “On the asymptotic limit of the three dimensional Vlasov–Poisson system for large magnetic field: Formal derivation”, J. Stat. Phys., 165:4 (2016), 765–784 | DOI | MR | Zbl

[9] Dinklage A., et al., “Magnetic configuration effects on the Wendelstein $7-X$ stellarator”, Nature Physics, 14:8 (2018), 855–860 | DOI

[10] Frénod E., Sonnendrücker E., “Homogenization of the Vlasov equation and of the Vlasov–Poisson system with a strong external magnetic field”, Asymptot. Anal., 18:3-4 (1998), 193–213 | MR

[11] Frieman E. A., Chen L., “Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria”, Phys. Fluids, 25:3 (1982), 502–508 | DOI | Zbl

[12] Golse F., Saint-Raymond L., “The Vlasov–Poisson system with strong magnetic field”, J. Math. Pures Appl. (9), 78:8 (1999), 791–817 | DOI | MR | Zbl

[13] Grad H., “Asymptotic theory of the Boltzmann equation”, Phys. Fluids, 6 (1963), 147–181 | DOI | MR | Zbl

[14] Hilbert D., “Begründung der kinetischen Gastheorie”, Math. Ann., 72:4 (1912), 562–577 | DOI | MR | Zbl

[15] Kessler T., Rjasanow S., Weisser S., “Vlasov–Poisson system tackled by particle simulation utilizing boundary element methods”, SIAM J. Sci. Comput., 42:1 (2020), B299–B326 | DOI | MR | Zbl

[16] Krommes J. A., “The gyrokinetic description of microturbulence in magnetized plasmas”, Annu. Rev. Fluid Mech., 44:1 (2012), 175–201 | DOI | MR | Zbl

[17] Pedersen T. S., et al., “First results from divertor operation in Wendelstein ${7-X}$”, Plasma Phys. Control. Fusion, 61:1 (2018), 014035 | DOI

[18] Possanner S., “Gyrokinetics from variational averaging: Existence and error bounds”, J. Math. Phys., 59:8 (2018), 082702 | DOI | MR | Zbl

[19] Shafranov V. D., Bondarenko B. D., Goncharov G. A., Lavrentev O. A., Sakharov A. D., “K istorii issledovanii po upravlyaemomu termoyadernomu sintezu”, Uspekhi fiz. nauk, 44:8 (2001), 835–843

[20] Sips A. C. C., et al., “Advanced scenarios for ITER operation”, Plasma Phys. Control. Fusion, 47:5A (2005), A19–A40 | DOI

[21] Vedenyapin V. V., “Granichnye zadachi dlya statsionarnogo uravneniya Vlasova”, Dokl. Akad. nauk SSSR, 20:4 (1986), 777–780

[22] Vedenyapin V. V., “O klassifikatsii statsionarnykh reshenii uravneniya Vlasova na tore i granichnaya zadacha”, Dokl. Akad. nauk SSSR, 323:6 (1992), 1004–1006 | Zbl

[23] Vedenyapin V. V., Sinitsyn A., Dulov E., Kinetic Boltzmann, Vlasov and related equations, Elsevier, Inc., Amsterdam, 2011 | MR | Zbl

[24] Vlasov A. A., “On vibration properties of electron gas”, J. Exp. Theor. Phys., 8:3 (1938), 291 | Zbl