August  2022, 15(4): 569-604. doi: 10.3934/krm.2021039

Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus

1. 

Department of Mathematics, Pohang University of Science and Technology, Pohang, Republic of Korea

2. 

Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104, USA

3. 

Department of Mathematics, The University of Hong Kong, Pokfulam, Hong Kong

* Corresponding author: Robert M. Strain

†Supported by the German DFG grant CRC 1060, the Korean Basic Science Research Institute Fund NRF-2021R1A6A1A10042944 and the Korean IBS grant IBS-R003-D1.
*Partially supported by the NSF grants DMS-1764177 and DMS-2055271 of the USA.
‡Partially supported by the HKU Seed Fund for Basic Research under the project code 201702159009, the Start-up Allowance for Croucher Award Recipients, Hong Kong General Research Fund (GRF) grant "Solving Generic Mean Field Type Problems: Interplay between Partial Differential Equations and Stochastic Analysis" with project number 17306420, and Hong Kong GRF grant "Controlling the Growth of Classical Solutions of a Class of Parabolic Differential Equations with Singular Coefficients: Resolutions for Some Lasting Problems from Economics" with project number 17302521.

Received  July 2021 Revised  November 2021 Published  August 2022 Early access  November 2021

Although the nuclear fusion process has received a great deal of attention in recent years, the amount of mathematical analysis that supports the stability of the system seems to be relatively insufficient. This paper deals with the mathematical analysis of the magnetic confinement of the plasma via kinetic equations. We prove the global wellposedness of the Vlasov-Maxwell system in a two-dimensional annulus when a huge (but finite-in-time) external magnetic potential is imposed near the boundary. We assume that the solution is axisymmetric. The authors hope that this work is a step towards a more generalized work on the three-dimensional Tokamak structure. The highlight of this work is the physical assumptions on the external magnetic potential well which remains finite within a finite time interval and from that, we prove that the plasma never touches the boundary. In addition, we provide a sufficient condition on the magnitude of the external magnetic potential to guarantee that the plasma is confined in an annulus of the desired thickness which is slightly larger than the initial support. Our method uses the cylindrical coordinate forms of the Vlasov-Maxwell system.

Citation: Jin Woo Jang, Robert M. Strain, Tak Kwong Wong. Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus. Kinetic and Related Models, 2022, 15 (4) : 569-604. doi: 10.3934/krm.2021039
References:
[1]

Y. O. BelyaevaB. Gebhard and A. L. Skubachevskii, A general way to confined stationary Vlasov-Poisson plasma configurations, Kinet. Relat. Models, 14 (2021), 257-282.  doi: 10.3934/krm.2021004.

[2]

F. BouchutF. Golse and C. Pallard, Classical solutions and the Glassey-Strauss theorem for the 3D Vlasov-Maxwell system, Arch. Ration. Mech. Anal., 170 (2003), 1-15.  doi: 10.1007/s00205-003-0265-6.

[3]

S. CaprinoG. Cavallaro and C. Marchioro, On a magnetically confined plasma with infinite charge, SIAM J. Math. Anal., 46 (2014), 133-164.  doi: 10.1137/130916527.

[4]

S. CaprinoG. Cavallaro and C. Marchioro, On a Vlasov-Poisson plasma confined in a torus by a magnetic mirror, J. Math. Anal. Appl., 427 (2015), 31-46.  doi: 10.1016/j.jmaa.2015.02.012.

[5]

S. CaprinoG. Cavallaro and C. Marchioro, On the magnetic shield for a Vlasov-Poisson plasma, J. Stat. Phys., 169 (2017), 1066-1097.  doi: 10.1007/s10955-017-1913-9.

[6]

S. CaprinoG. Cavallaro and C. Marchioro, Time evolution of a Vlasov-Poisson plasma with magnetic confinement, Kinet. Relat. Models, 5 (2012), 729-742.  doi: 10.3934/krm.2012.5.729.

[7]

G. A. Cottrell and R. O. Dendy, Superthermal radiation from fusion products in JET, Phys. Rev. Lett., 60 (1988). doi: 10.1103/PhysRevLett.60.33.

[8]

F. W. Crawford, A review of cyclotron harmonic phenomena in plasmas, Nuclear Fusion, 5 (1965). doi: 10.1088/0029-5515/5/1/010.

[9]

P. Degond, Local existence of solutions of the Vlasov-Maxwell equations and convergence to the Vlasov-Poisson equations for infinite light velocity, Math. Methods Appl. Sci., 8 (1986), 533-558.  doi: 10.1002/mma.1670080135.

[10]

P. Degond and F. Filbet, On the asymptotic limit of the three dimensional Vlasov-Poisson system for large magnetic field: Formal derivation, J. Stat. Phys., 165 (2016), 765-784.  doi: 10.1007/s10955-016-1645-2.

[11]

R. J. DiPerna and P.-L. Lions, Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math., 42 (1989), 729-757.  doi: 10.1002/cpa.3160420603.

[12]

R. L. Dobrushin, Vlasov equations, Funct. Anal. Appl., 13 (1979), 115-123.  doi: 10.1007/BF01077243.

[13]

A. FasoliS. BrunnerW. A. CooperJ. P. GravesP. RicciO. Sauter and L. Villard, Computational challenges in magnetic-confinement fusion physics, Nature Physics, 12 (2016), 411-423.  doi: 10.1038/nphys3744.

[14]

F. Filbet and L. M. Rodrigues, Asymptotically preserving particle-in-cell methods for inhomogeneous strongly magnetized plasmas, SIAM J. Numer. Anal., 55 (2017), 2416-2443.  doi: 10.1137/17M1113229.

[15]

F. Filbet and L. M. Rodrigues, Asymptotically stable particle-in-cell methods for the Vlasov-Poisson system with a strong external magnetic field, SIAM J. Numer. Anal., 54 (2016), 1120-1146.  doi: 10.1137/15M104952X.

[16]

F. Filbet and L. M. Rodrigues, Asymptotics of the three-dimensional Vlasov equation in the large magnetic field limit, J. Éc. Polytech. Math., 7 (2020), 1009-1067.  doi: 10.5802/jep.134.

[17]

F. FilbetT. Xiong and E. Sonnendrücker, On the Vlasov-Maxwell system with a strong magnetic field, SIAM J. Appl. Math., 78 (2018), 1030-1055.  doi: 10.1137/17M1112030.

[18]

P. R. Garabedian, A unified theory of tokamaks and stellarators, Comm. Pure Appl. Math., 47 (1994), 281-292.  doi: 10.1002/cpa.3160470303.

[19]

R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477.

[20]

R. T. Glassey and J. Schaeffer, On the "one and one-half dimensional'' relativistic Vlasov-Maxwell system, Math. Methods Appl. Sci., 13 (1990), 169-179.  doi: 10.1002/mma.1670130207.

[21]

R. T. Glassey and J. Schaeffer, The relativistic Vlasov-Maxwell system in 2D and 2.5D, in Nonlinear Wave Equations, Contemp. Math., 263, Amer. Math. Soc., Providence, RI, 2000, 61–69. doi: 10.1090/conm/263/04191.

[22]

R. T. Glassey and J. Schaeffer, The relativistic Vlasov-Maxwell system in two space dimensions. I, II, Arch. Rational Mech. Anal., 141 (1998), 331–354,355–374. doi: 10.1007/s002050050079.

[23]

R. T. Glassey and J. Schaeffer, The "two and one-half-dimensional'' relativistic Vlasov Maxwell system, Comm. Math. Phys., 185 (1997), 257-284.  doi: 10.1007/s002200050090.

[24]

R. T. Glassey and J. W. Schaeffer, Global existence for the relativistic Vlasov-Maxwell system with nearly neutral initial data, Comm. Math. Phys., 119 (1988), 353-384.  doi: 10.1007/BF01218078.

[25]

R. T. Glassey and W. A. Strauss, High velocity particles in a collisionless plasma, Math. Methods Appl. Sci., 9 (1987), 46-52.  doi: 10.1002/mma.1670090105.

[26]

R. T. Glassey and W. A. Strauss, Remarks on collisionless plasmas, in Fluids and Plasmas: Geometry and Dynamics, Contemp. Math., 28, Amer. Math. Soc., Providence, RI, 1984,269–279. doi: 10.1090/conm/028/751989.

[27]

R. T. Glassey and W. A. Strauss, Singularity formation in a collisionless plasma could occur only at high velocities, Arch. Rational Mech. Anal., 92 (1986), 59-90.  doi: 10.1007/BF00250732.

[28]

A. P. H. Goede, P. Massmann, H. J. Hopman and J. Kistemaker, Ion Bernstein waves excited by an energeticion beam ion a plasma, Nuclear Fusion, 16 (1976). doi: 10.1088/0029-5515/16/1/009.

[29]

G. Guest, Electron Cyclotron Heating of Plasmas, Vol. 3, Wiley Online Library, 2009.

[30]

Y. Guo, Global weak solutions of the Vlasov-Maxwell system with boundary conditions, Comm. Math. Phys., 154 (1993), 245-263.  doi: 10.1007/BF02096997.

[31]

Y. Guo, Singular solutions of the Vlasov-Maxwell system on a half line, Arch. Rational Mech. Anal., 131 (1995), 241-304.  doi: 10.1007/BF00382888.

[32]

Y. Guo, Stable magnetic equilibria in collisionless plasmas, Comm. Pure Appl. Math., 50 (1997), 891-933.  doi: 10.1002/(SICI)1097-0312(199709)50:9<891::AID-CPA4>3.0.CO;2-0.

[33]

Y. Guo, Stable magnetic equilibria in a symmetric collisionless plasma, Comm. Math. Phys., 200 (1999), 211-247.  doi: 10.1007/s002200050528.

[34]

D. Han-Kwan, On the confinement of a tokamak plasma, SIAM J. Math. Anal., 42 (2010), 2337-2367.  doi: 10.1137/090774574.

[35]

E. Horst, Global solutions of the relativistic Vlasov-Maxwell system of plasma physics, Dissertationes Math. (Rozprawy Mat.), 292 (1990), 63pp.

[36]

R. F. Hubbard and T. J. Birmingham, Electrostatic emissions between electron gyroharmonics in the outer magnetosphere, J. Geophys. Res.: Space Physics, 83 (1978), 4837-4850.  doi: 10.1029/JA083iA10p04837.

[37]

M. Kunze, Yet another criterion for global existence in the 3D relativistic Vlasov-Maxwell system, J. Differential Equations, 259 (2015), 4413-4442.  doi: 10.1016/j.jde.2015.06.003.

[38]

J. Luk and R. M. Strain, A new continuation criterion for the relativistic Vlasov-Maxwell system, Comm. Math. Phys., 331 (2014), 1005-1027.  doi: 10.1007/s00220-014-2108-8.

[39]

J. Luk and R. M. Strain, Strichartz estimates and moment bounds for the relativistic Vlasov-Maxwell system, Arch. Ration. Mech. Anal., 219 (2016), 445-552.  doi: 10.1007/s00205-015-0899-1.

[40]

T. T. NguyenT. V. Nguyen and W. A. Strauss, Erratum to: Global magnetic confinement for the 1.5D Vlasov-Maxwell system, Kinet. Relat. Models, 8 (2015), 615-616.  doi: 10.3934/krm.2015.8.615.

[41]

T. T. NguyenT. V. Nguyen and W. A. Strauss, Global magnetic confinement for the 1.5D Vlasov-Maxwell system, Kinet. Relat. Models, 8 (2015), 153-168.  doi: 10.3934/krm.2015.8.153.

[42]

J. OngenaR. KochR. Wolf and H. Zohm, Magnetic-confinement fusion, Nature Physics, 12 (2016), 398-410.  doi: 10.1038/nphys3745.

[43]

N. Patel, Three new results on continuation criteria for the 3D relativistic Vlasov-Maxwell system, J. Differential Equations, 264 (2018), 1841-1885.  doi: 10.1016/j.jde.2017.10.008.

[44]

S. PerrautA. RouxP. RobertR. GendrinJ.-A. SauvaudJ.-M. BosquedG. Kremser and A. Korth, A systematic study of ULF waves above $F_{H+}$ from GEOS 1 and 2 measurements and their relationships with proton ring distributions, J. Geophys. Res.: Space Physics, 87 (1982), 6219-6236.  doi: 10.1029/JA087iA08p06219.

[45]

R. F. Post and M. N. Rosenbluth, Electrostatic instabilities in finite mirror-confined plasmas, Phys. Fluids, 9 (1966), 730-749.  doi: 10.1063/1.1761740.

[46]

G. Rein, Generic global solutions of the relativistic Vlasov-Maxwell system of plasma physics, Comm. Math. Phys., 135 (1990), 41-78.  doi: 10.1007/BF02097656.

[47]

R. M. Strain, The Vlasov-Maxwell-Boltzmann system in the whole space, Comm. Math. Phys., 268 (2006), 543-567.  doi: 10.1007/s00220-006-0109-y.

[48]

H. Tasso and G. Throumoulopoulos, Tokamak-like Vlasov equilibria, European Phys. J. D, 68 (2014). doi: 10.1140/epjd/e2014-50007-9.

[49]

G. Vogman, Fourth-Order Conservative Vlasov-Maxwell Solver for Cartesian and Cylindrical Phase Space Coordinates, Ph.D thesis, University of California in Berkeley, 2016.

[50]

R. B. White, The Theory of Toroidally Confined Plasmas, 2nd edition, Imperial College Press, London, 2001. doi: 10.1142/p237.

show all references

References:
[1]

Y. O. BelyaevaB. Gebhard and A. L. Skubachevskii, A general way to confined stationary Vlasov-Poisson plasma configurations, Kinet. Relat. Models, 14 (2021), 257-282.  doi: 10.3934/krm.2021004.

[2]

F. BouchutF. Golse and C. Pallard, Classical solutions and the Glassey-Strauss theorem for the 3D Vlasov-Maxwell system, Arch. Ration. Mech. Anal., 170 (2003), 1-15.  doi: 10.1007/s00205-003-0265-6.

[3]

S. CaprinoG. Cavallaro and C. Marchioro, On a magnetically confined plasma with infinite charge, SIAM J. Math. Anal., 46 (2014), 133-164.  doi: 10.1137/130916527.

[4]

S. CaprinoG. Cavallaro and C. Marchioro, On a Vlasov-Poisson plasma confined in a torus by a magnetic mirror, J. Math. Anal. Appl., 427 (2015), 31-46.  doi: 10.1016/j.jmaa.2015.02.012.

[5]

S. CaprinoG. Cavallaro and C. Marchioro, On the magnetic shield for a Vlasov-Poisson plasma, J. Stat. Phys., 169 (2017), 1066-1097.  doi: 10.1007/s10955-017-1913-9.

[6]

S. CaprinoG. Cavallaro and C. Marchioro, Time evolution of a Vlasov-Poisson plasma with magnetic confinement, Kinet. Relat. Models, 5 (2012), 729-742.  doi: 10.3934/krm.2012.5.729.

[7]

G. A. Cottrell and R. O. Dendy, Superthermal radiation from fusion products in JET, Phys. Rev. Lett., 60 (1988). doi: 10.1103/PhysRevLett.60.33.

[8]

F. W. Crawford, A review of cyclotron harmonic phenomena in plasmas, Nuclear Fusion, 5 (1965). doi: 10.1088/0029-5515/5/1/010.

[9]

P. Degond, Local existence of solutions of the Vlasov-Maxwell equations and convergence to the Vlasov-Poisson equations for infinite light velocity, Math. Methods Appl. Sci., 8 (1986), 533-558.  doi: 10.1002/mma.1670080135.

[10]

P. Degond and F. Filbet, On the asymptotic limit of the three dimensional Vlasov-Poisson system for large magnetic field: Formal derivation, J. Stat. Phys., 165 (2016), 765-784.  doi: 10.1007/s10955-016-1645-2.

[11]

R. J. DiPerna and P.-L. Lions, Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math., 42 (1989), 729-757.  doi: 10.1002/cpa.3160420603.

[12]

R. L. Dobrushin, Vlasov equations, Funct. Anal. Appl., 13 (1979), 115-123.  doi: 10.1007/BF01077243.

[13]

A. FasoliS. BrunnerW. A. CooperJ. P. GravesP. RicciO. Sauter and L. Villard, Computational challenges in magnetic-confinement fusion physics, Nature Physics, 12 (2016), 411-423.  doi: 10.1038/nphys3744.

[14]

F. Filbet and L. M. Rodrigues, Asymptotically preserving particle-in-cell methods for inhomogeneous strongly magnetized plasmas, SIAM J. Numer. Anal., 55 (2017), 2416-2443.  doi: 10.1137/17M1113229.

[15]

F. Filbet and L. M. Rodrigues, Asymptotically stable particle-in-cell methods for the Vlasov-Poisson system with a strong external magnetic field, SIAM J. Numer. Anal., 54 (2016), 1120-1146.  doi: 10.1137/15M104952X.

[16]

F. Filbet and L. M. Rodrigues, Asymptotics of the three-dimensional Vlasov equation in the large magnetic field limit, J. Éc. Polytech. Math., 7 (2020), 1009-1067.  doi: 10.5802/jep.134.

[17]

F. FilbetT. Xiong and E. Sonnendrücker, On the Vlasov-Maxwell system with a strong magnetic field, SIAM J. Appl. Math., 78 (2018), 1030-1055.  doi: 10.1137/17M1112030.

[18]

P. R. Garabedian, A unified theory of tokamaks and stellarators, Comm. Pure Appl. Math., 47 (1994), 281-292.  doi: 10.1002/cpa.3160470303.

[19]

R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477.

[20]

R. T. Glassey and J. Schaeffer, On the "one and one-half dimensional'' relativistic Vlasov-Maxwell system, Math. Methods Appl. Sci., 13 (1990), 169-179.  doi: 10.1002/mma.1670130207.

[21]

R. T. Glassey and J. Schaeffer, The relativistic Vlasov-Maxwell system in 2D and 2.5D, in Nonlinear Wave Equations, Contemp. Math., 263, Amer. Math. Soc., Providence, RI, 2000, 61–69. doi: 10.1090/conm/263/04191.

[22]

R. T. Glassey and J. Schaeffer, The relativistic Vlasov-Maxwell system in two space dimensions. I, II, Arch. Rational Mech. Anal., 141 (1998), 331–354,355–374. doi: 10.1007/s002050050079.

[23]

R. T. Glassey and J. Schaeffer, The "two and one-half-dimensional'' relativistic Vlasov Maxwell system, Comm. Math. Phys., 185 (1997), 257-284.  doi: 10.1007/s002200050090.

[24]

R. T. Glassey and J. W. Schaeffer, Global existence for the relativistic Vlasov-Maxwell system with nearly neutral initial data, Comm. Math. Phys., 119 (1988), 353-384.  doi: 10.1007/BF01218078.

[25]

R. T. Glassey and W. A. Strauss, High velocity particles in a collisionless plasma, Math. Methods Appl. Sci., 9 (1987), 46-52.  doi: 10.1002/mma.1670090105.

[26]

R. T. Glassey and W. A. Strauss, Remarks on collisionless plasmas, in Fluids and Plasmas: Geometry and Dynamics, Contemp. Math., 28, Amer. Math. Soc., Providence, RI, 1984,269–279. doi: 10.1090/conm/028/751989.

[27]

R. T. Glassey and W. A. Strauss, Singularity formation in a collisionless plasma could occur only at high velocities, Arch. Rational Mech. Anal., 92 (1986), 59-90.  doi: 10.1007/BF00250732.

[28]

A. P. H. Goede, P. Massmann, H. J. Hopman and J. Kistemaker, Ion Bernstein waves excited by an energeticion beam ion a plasma, Nuclear Fusion, 16 (1976). doi: 10.1088/0029-5515/16/1/009.

[29]

G. Guest, Electron Cyclotron Heating of Plasmas, Vol. 3, Wiley Online Library, 2009.

[30]

Y. Guo, Global weak solutions of the Vlasov-Maxwell system with boundary conditions, Comm. Math. Phys., 154 (1993), 245-263.  doi: 10.1007/BF02096997.

[31]

Y. Guo, Singular solutions of the Vlasov-Maxwell system on a half line, Arch. Rational Mech. Anal., 131 (1995), 241-304.  doi: 10.1007/BF00382888.

[32]

Y. Guo, Stable magnetic equilibria in collisionless plasmas, Comm. Pure Appl. Math., 50 (1997), 891-933.  doi: 10.1002/(SICI)1097-0312(199709)50:9<891::AID-CPA4>3.0.CO;2-0.

[33]

Y. Guo, Stable magnetic equilibria in a symmetric collisionless plasma, Comm. Math. Phys., 200 (1999), 211-247.  doi: 10.1007/s002200050528.

[34]

D. Han-Kwan, On the confinement of a tokamak plasma, SIAM J. Math. Anal., 42 (2010), 2337-2367.  doi: 10.1137/090774574.

[35]

E. Horst, Global solutions of the relativistic Vlasov-Maxwell system of plasma physics, Dissertationes Math. (Rozprawy Mat.), 292 (1990), 63pp.

[36]

R. F. Hubbard and T. J. Birmingham, Electrostatic emissions between electron gyroharmonics in the outer magnetosphere, J. Geophys. Res.: Space Physics, 83 (1978), 4837-4850.  doi: 10.1029/JA083iA10p04837.

[37]

M. Kunze, Yet another criterion for global existence in the 3D relativistic Vlasov-Maxwell system, J. Differential Equations, 259 (2015), 4413-4442.  doi: 10.1016/j.jde.2015.06.003.

[38]

J. Luk and R. M. Strain, A new continuation criterion for the relativistic Vlasov-Maxwell system, Comm. Math. Phys., 331 (2014), 1005-1027.  doi: 10.1007/s00220-014-2108-8.

[39]

J. Luk and R. M. Strain, Strichartz estimates and moment bounds for the relativistic Vlasov-Maxwell system, Arch. Ration. Mech. Anal., 219 (2016), 445-552.  doi: 10.1007/s00205-015-0899-1.

[40]

T. T. NguyenT. V. Nguyen and W. A. Strauss, Erratum to: Global magnetic confinement for the 1.5D Vlasov-Maxwell system, Kinet. Relat. Models, 8 (2015), 615-616.  doi: 10.3934/krm.2015.8.615.

[41]

T. T. NguyenT. V. Nguyen and W. A. Strauss, Global magnetic confinement for the 1.5D Vlasov-Maxwell system, Kinet. Relat. Models, 8 (2015), 153-168.  doi: 10.3934/krm.2015.8.153.

[42]

J. OngenaR. KochR. Wolf and H. Zohm, Magnetic-confinement fusion, Nature Physics, 12 (2016), 398-410.  doi: 10.1038/nphys3745.

[43]

N. Patel, Three new results on continuation criteria for the 3D relativistic Vlasov-Maxwell system, J. Differential Equations, 264 (2018), 1841-1885.  doi: 10.1016/j.jde.2017.10.008.

[44]

S. PerrautA. RouxP. RobertR. GendrinJ.-A. SauvaudJ.-M. BosquedG. Kremser and A. Korth, A systematic study of ULF waves above $F_{H+}$ from GEOS 1 and 2 measurements and their relationships with proton ring distributions, J. Geophys. Res.: Space Physics, 87 (1982), 6219-6236.  doi: 10.1029/JA087iA08p06219.

[45]

R. F. Post and M. N. Rosenbluth, Electrostatic instabilities in finite mirror-confined plasmas, Phys. Fluids, 9 (1966), 730-749.  doi: 10.1063/1.1761740.

[46]

G. Rein, Generic global solutions of the relativistic Vlasov-Maxwell system of plasma physics, Comm. Math. Phys., 135 (1990), 41-78.  doi: 10.1007/BF02097656.

[47]

R. M. Strain, The Vlasov-Maxwell-Boltzmann system in the whole space, Comm. Math. Phys., 268 (2006), 543-567.  doi: 10.1007/s00220-006-0109-y.

[48]

H. Tasso and G. Throumoulopoulos, Tokamak-like Vlasov equilibria, European Phys. J. D, 68 (2014). doi: 10.1140/epjd/e2014-50007-9.

[49]

G. Vogman, Fourth-Order Conservative Vlasov-Maxwell Solver for Cartesian and Cylindrical Phase Space Coordinates, Ph.D thesis, University of California in Berkeley, 2016.

[50]

R. B. White, The Theory of Toroidally Confined Plasmas, 2nd edition, Imperial College Press, London, 2001. doi: 10.1142/p237.

Figure 1.  The domain $ \Delta $
[1]

Silvia Caprino, Guido Cavallaro, Carlo Marchioro. Time evolution of a Vlasov-Poisson plasma with magnetic confinement. Kinetic and Related Models, 2012, 5 (4) : 729-742. doi: 10.3934/krm.2012.5.729

[2]

Toan T. Nguyen, Truyen V. Nguyen, Walter A. Strauss. Global magnetic confinement for the 1.5D Vlasov-Maxwell system. Kinetic and Related Models, 2015, 8 (1) : 153-168. doi: 10.3934/krm.2015.8.153

[3]

Toan T. Nguyen, Truyen V. Nguyen, Walter A. Strauss. Erratum to: Global magnetic confinement for the 1.5D Vlasov-Maxwell system. Kinetic and Related Models, 2015, 8 (3) : 615-616. doi: 10.3934/krm.2015.8.615

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