Limit model for the Vlasov–Maxwell system with strong magnetic fields via gyroaveraging

Keßler, T.; Rjasanow, S.

doi:10.1090/spmj/1668

Limit model for the Vlasov–Maxwell system with strong magnetic fields via gyroaveraging
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by T. Keßler and S. Rjasanow

St. Petersburg Math. J. 32 (2021), 753-765

DOI: https://doi.org/10.1090/spmj/1668

Published electronically: July 9, 2021

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Abstract:

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.

References

Yu. O. Belyaeva, Stationary solutions of Vlasov equations for a high-temperature two-component plasma, Sovrem. Mat. Fundam. Napravl. 62 (2016), 19–31 (Russian, with Russian summary); English transl., J. Math. Sci. (N.Y.) 239 (2019), no. 6, 739–750. MR 3611728
Yu. O. Belyaeva and A. L. Skubachevskiĭ, On the unique solvability of the first mixed problem for the Vlasov-Poisson system of equations in an infinite cylinder, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 477 (2018), no. Kraevye Zadachi Matematicheskoĭ Fiziki i Smezhnye Voprosy Teorii Funktsiĭ. 47, 12–34 (Russian, with English summary); English transl., J. Math. Sci. (N.Y.) 244 (2020), no. 6, 930–945. MR 3904054
A. H. Boozer, Physics of magnetically confined plasmas, Rev. Mod. Phys. 76 (2005), 1071–1141.
Mihai Bostan, Gyrokinetic Vlasov equation in three dimensional setting. Second order approximation, Multiscale Model. Simul. 8 (2010), no. 5, 1923–1957. MR 2769087, DOI 10.1137/090777621
A. J. Brizard and T. S. Hahm, Foundations of nonlinear gyrokinetic theory, Rev. Modern Phys. 79 (2007), no. 2, 421–468. MR 2336960, DOI 10.1103/RevModPhys.79.421
Carlo Cercignani, The Boltzmann equation and its applications, Applied Mathematical Sciences, vol. 67, Springer-Verlag, New York, 1988. MR 1313028, DOI 10.1007/978-1-4612-1039-9
F. F. Chen, Introduction to plasma physics and controlled fusion, Springer Int. Publ., 2016.
Pierre Degond and Francis Filbet, On the asymptotic limit of the three dimensional Vlasov-Poisson system for large magnetic field: formal derivation, J. Stat. Phys. 165 (2016), no. 4, 765–784. MR 3568167, DOI 10.1007/s10955-016-1645-2
A. Dinklage, et al., Magnetic configuration effects on the Wendelstein $7-X$ stellarator, Nature Physics 14 (2018), no. 8, 855–860.
Emmanuel Frénod and Eric Sonnendrücker, Homogenization of the Vlasov equation and of the Vlasov-Poisson system with a strong external magnetic field, Asymptot. Anal. 18 (1998), no. 3-4, 193–213. MR 1668938
E. A. Frieman and L. Chen, Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria, Phys. Fluids 25 (1982), no. 3, 502–508.
François Golse and Laure Saint-Raymond, The Vlasov-Poisson system with strong magnetic field, J. Math. Pures Appl. (9) 78 (1999), no. 8, 791–817. MR 1715342, DOI 10.1016/S0021-7824(99)00021-5
Harold Grad, Asymptotic theory of the Boltzmann equation, Phys. Fluids 6 (1963), 147–181. MR 155541, DOI 10.1063/1.1706716
David Hilbert, Begründung der kinetischen Gastheorie, Math. Ann. 72 (1912), no. 4, 562–577 (German). MR 1511713, DOI 10.1007/BF01456676
Torsten Keßler, Sergej Rjasanow, and Steffen Weißer, Vlasov-Poisson system tackled by particle simulation utilizing boundary element methods, SIAM J. Sci. Comput. 42 (2020), no. 1, B299–B326. MR 4065197, DOI 10.1137/18M1225823
John A. Krommes, The gyrokinetic description of microturbulence in magnetized plasmas, Annual review of fluid mechanics. Volume 44, 2012, Annu. Rev. Fluid Mech., vol. 44, Annual Reviews, Palo Alto, CA, 2012, pp. 175–201. MR 2882594, DOI 10.1146/annurev-fluid-120710-101223
T. S. Pedersen, et al., First results from divertor operation in Wendelstein ${7-X}$, Plasma Phys. Control. Fusion 61 (2018), no. 1, 014035.
Stefan Possanner, Gyrokinetics from variational averaging: existence and error bounds, J. Math. Phys. 59 (2018), no. 8, 082702, 34. MR 3843633, DOI 10.1063/1.5018354
V. D. Shafranov, B. D. Bondarenko, G. A. Goncharov, O. A. Lavrent′ev, and A. D. Sakharov, On the history of the research into controlled thermonuclear fusion, Uspekhi Phis. Nauk 171 (2001), 877–886; English transl., Phys. Usp. 44 (2001), no. 8, 835–843.
A. C. C. Sips, et al., Advanced scenarios for ITER operation, Plasma Phys. Control. Fusion 47 (2005), no. 5A, A19–A40.
V. V. Vedenyapin, Boundary value problems for a stationary Vlasov equation, Dokl. Akad. Nauk SSSR 290 (1986), no. 4, 777–780 (Russian). MR 863352
V. V. Vedenyapin, Classification of stationary solutions of the Vlasov equation on a torus and a boundary value problem, Dokl. Akad. Nauk 323 (1992), no. 6, 1004–1006 (Russian); English transl., Russian Acad. Sci. Dokl. Math. 45 (1992), no. 2, 459–462 (1993). MR 1202299
Victor Vedenyapin, Alexander Sinitsyn, and Eugene Dulov, Kinetic Boltzmann, Vlasov and related equations, Elsevier, Inc., Amsterdam, 2011. MR 3294571, DOI 10.1016/B978-0-12-387779-6.00001-6
A. A. Vlasov, On vibration properties of electron gas, J. Exp. Theor. Phys. 8 (1938), no. 3, 291.