December  2020, 13(6): 1135-1161. doi: 10.3934/krm.2020040

Confined steady states of the relativistic Vlasov–Maxwell system in an infinitely long cylinder

University of Bayreuth, Universitätsstraße 30, 95440 Bayreuth, Germany

Received  March 2020 Revised  June 2020 Published  December 2020 Early access  September 2020

The time evolution of a collisionless plasma is modeled by the relativistic Vlasov–Maxwell system which couples the Vlasov equation (the transport equation) with the Maxwell equations of electrodynamics. In this work, the setting is two and one-half dimensional, that is, the distribution functions of the particles species are independent of the third space dimension. We consider the case that the plasma is located in an infinitely long cylinder and is influenced by an external magnetic field. We prove existence of stationary solutions and give conditions on the external magnetic field under which the plasma is confined inside the cylinder, i.e., it stays away from the boundary of the cylinder.

Citation: Jörg Weber. Confined steady states of the relativistic Vlasov–Maxwell system in an infinitely long cylinder. Kinetic and Related Models, 2020, 13 (6) : 1135-1161. doi: 10.3934/krm.2020040
References:
[1]

J. Batt and K. Fabian, Stationary solutions of the relativistic Vlasov–Maxwell system of plasma physics, Chin. Ann. Math. Ser. B, 14 (1993), 253-278. 

[2]

P. R. Beesack, Comparison theorems and integral inequalities for Volterra integral equations, Proc. Amer. Math. Soc., 20 (1969), 61-66.  doi: 10.1090/S0002-9939-1969-0234228-3.

[3]

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.

[4]

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.

[5]

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.

[6]

S. CaprinoG. Cavallaro and C. Marchioro, A Vlasov–Poisson plasma with unbounded mass and velocities confined in a cylinder by a magnetic mirror, Kinet. Relat. Models, 9 (2016), 657-686.  doi: 10.3934/krm.2016011.

[7]

P. Degond, Solutions stationnaires explicites du système de Vlasov–Maxwell relativiste, C. R. Math. Acad. Sci. Paris, 310 (1990), 607-612. 

[8]

L. C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics, American Mathematical Society, 2 edition, 2010. doi: 10.1090/gsm/019.

[9]

R. 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.

[10]

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

[11]

P. Knopf, Optimal control of a Vlasov–Poisson plasma by an external magnetic field, Calc. Var. Partial Differential Equations, 57 (2018), No. 134, 37 pp. doi: 10.1007/s00526-018-1407-x.

[12]

P. Knopf, Confined steady states of a Vlasov–Poisson plasma in an infinitely long cylinder, Math. Methods Appl. Sci., 42 (2019), 6369-6384.  doi: 10.1002/mma.5728.

[13]

P. Knopf and J. Weber, Optimal control of a Vlasov–Poisson plasma by fixed magnetic field coils, Appl. Math. Optim., 81 (2020), 961-988.  doi: 10.1007/s00245-018-9526-5.

[14]

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.

[15]

T. T. Nguyen and W. A. Strauss, Linear stability analysis of a hot plasma in a solid torus, Arch. Ration. Mech. Anal., 211 (2014), 619-672.  doi: 10.1007/s00205-013-0680-2.

[16]

F. Poupaud, Boundary value problems for the stationary Vlasov–Maxwell system, Forum Math., 4 (1992), 499-527.  doi: 10.1515/form.1992.4.499.

[17]

G. Rein, Existence of stationary, collisionless plasmas in bounded domains, Math. Methods Appl. Sci., 15 (1992), 365-374.  doi: 10.1002/mma.1670150507.

[18]

A. L. Skubachevskii, Vlasov–Poisson equations for a two-component plasma in a homogeneous magnetic field, Russian Math. Surveys, 69 (2014), 291-330.  doi: 10.1070/rm2014v069n02abeh004889.

[19]

W. Stacey, Fusion Plasma Physics, Physics textbook. Wiley-VCH, 2 edition, 2012. doi: 10.1002/9783527669516.

[20]

F. G. Tricomi, Integral Equations, volume 5 of Pure and Applied Mathematics, Interscience Publishers, 1957.

[21]

J. Weber, Hot plasma in a container–-an optimal control problem, SIAM J. Math. Anal., 52 (2020), 2895-2929.  doi: 10.1137/19M1275061.

[22]

J. Weber, Optimal control of the two-dimensional Vlasov–Maxwell system, arXiv e-prints, arXiv: 1809.10016.

[23]

J. Weber, Weak solutions of the relativistic Vlasov–Maxwell system with external currents, arXiv e-prints, arXiv: 1902.02712.

[24]

K. Z. Zhang, Linear stability analysis of the relativistic Vlasov–Maxwell system in an axisymmetric domain, SIAM J. Math. Anal., 51 (2019), 4683-4723.  doi: 10.1137/18M1206825.

[25]

D. ZhelyazovD. Han-Kwan and J. D. M. Rademacher, Global stability and local bifurcations in a two-fluid model for Tokamak plasma, SIAM J. Appl. Dyn. Syst., 14 (2015), 730-763.  doi: 10.1137/130912384.

show all references

References:
[1]

J. Batt and K. Fabian, Stationary solutions of the relativistic Vlasov–Maxwell system of plasma physics, Chin. Ann. Math. Ser. B, 14 (1993), 253-278. 

[2]

P. R. Beesack, Comparison theorems and integral inequalities for Volterra integral equations, Proc. Amer. Math. Soc., 20 (1969), 61-66.  doi: 10.1090/S0002-9939-1969-0234228-3.

[3]

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.

[4]

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.

[5]

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.

[6]

S. CaprinoG. Cavallaro and C. Marchioro, A Vlasov–Poisson plasma with unbounded mass and velocities confined in a cylinder by a magnetic mirror, Kinet. Relat. Models, 9 (2016), 657-686.  doi: 10.3934/krm.2016011.

[7]

P. Degond, Solutions stationnaires explicites du système de Vlasov–Maxwell relativiste, C. R. Math. Acad. Sci. Paris, 310 (1990), 607-612. 

[8]

L. C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics, American Mathematical Society, 2 edition, 2010. doi: 10.1090/gsm/019.

[9]

R. 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.

[10]

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

[11]

P. Knopf, Optimal control of a Vlasov–Poisson plasma by an external magnetic field, Calc. Var. Partial Differential Equations, 57 (2018), No. 134, 37 pp. doi: 10.1007/s00526-018-1407-x.

[12]

P. Knopf, Confined steady states of a Vlasov–Poisson plasma in an infinitely long cylinder, Math. Methods Appl. Sci., 42 (2019), 6369-6384.  doi: 10.1002/mma.5728.

[13]

P. Knopf and J. Weber, Optimal control of a Vlasov–Poisson plasma by fixed magnetic field coils, Appl. Math. Optim., 81 (2020), 961-988.  doi: 10.1007/s00245-018-9526-5.

[14]

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.

[15]

T. T. Nguyen and W. A. Strauss, Linear stability analysis of a hot plasma in a solid torus, Arch. Ration. Mech. Anal., 211 (2014), 619-672.  doi: 10.1007/s00205-013-0680-2.

[16]

F. Poupaud, Boundary value problems for the stationary Vlasov–Maxwell system, Forum Math., 4 (1992), 499-527.  doi: 10.1515/form.1992.4.499.

[17]

G. Rein, Existence of stationary, collisionless plasmas in bounded domains, Math. Methods Appl. Sci., 15 (1992), 365-374.  doi: 10.1002/mma.1670150507.

[18]

A. L. Skubachevskii, Vlasov–Poisson equations for a two-component plasma in a homogeneous magnetic field, Russian Math. Surveys, 69 (2014), 291-330.  doi: 10.1070/rm2014v069n02abeh004889.

[19]

W. Stacey, Fusion Plasma Physics, Physics textbook. Wiley-VCH, 2 edition, 2012. doi: 10.1002/9783527669516.

[20]

F. G. Tricomi, Integral Equations, volume 5 of Pure and Applied Mathematics, Interscience Publishers, 1957.

[21]

J. Weber, Hot plasma in a container–-an optimal control problem, SIAM J. Math. Anal., 52 (2020), 2895-2929.  doi: 10.1137/19M1275061.

[22]

J. Weber, Optimal control of the two-dimensional Vlasov–Maxwell system, arXiv e-prints, arXiv: 1809.10016.

[23]

J. Weber, Weak solutions of the relativistic Vlasov–Maxwell system with external currents, arXiv e-prints, arXiv: 1902.02712.

[24]

K. Z. Zhang, Linear stability analysis of the relativistic Vlasov–Maxwell system in an axisymmetric domain, SIAM J. Math. Anal., 51 (2019), 4683-4723.  doi: 10.1137/18M1206825.

[25]

D. ZhelyazovD. Han-Kwan and J. D. M. Rademacher, Global stability and local bifurcations in a two-fluid model for Tokamak plasma, SIAM J. Appl. Dyn. Syst., 14 (2015), 730-763.  doi: 10.1137/130912384.

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