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  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 & 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[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.  Google Scholar

[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.  Google Scholar

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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[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.  Google Scholar

[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.  Google Scholar

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