April  2021, 14(2): 257-282. doi: 10.3934/krm.2021004

A general way to confined stationary Vlasov-Poisson plasma configurations

1. 

RUDN University, 6, Mikluhko–Maklaya str., 117198 Moscow, Russia

2. 

Universität Leipzig, Augustusplatz 10, 04109 Leipzig, Germany

* corresponding author

Received  October 2020 Published  December 2020

Fund Project: The first and third author are supported by the Russian Foundation for Basic Research grant 20-01-00288 (section 5) and "RUDN University Program 5-100" (section 6). The second author is supported by the German-Russian Interdisciplinary Science Center (G-RISC) funded by the German Federal Foreign Office via the German Academic Exchange Service (DAAD) (Project numbers: M-2018b-2, A-2019b-5_d)

We address the existence of stationary solutions of the Vlasov-Poisson system on a domain $ \Omega\subset\mathbb{R}^3 $ describing a high-temperature plasma which due to the influence of an external magnetic field is spatially confined to a subregion of $ \Omega $. In a first part we provide such an existence result for a generalized system of Vlasov-Poisson type and investigate the relation between the strength of the external magnetic field, the sharpness of the confinement and the amount of plasma that is confined measured in terms of the total charges. The key tools here are the method of sub-/supersolutions and the use of first integrals in combination with cutoff functions. In a second part we apply these general results to the usual Vlasov-Poisson equation in three different settings: the infinite and finite cylinder, as well as domains with toroidal symmetry. This way we prove the existence of stationary solutions corresponding to a two-component plasma confined in a Mirror trap, as well as a Tokamak.

Citation: Yulia O. Belyaeva, Björn Gebhard, Alexander L. Skubachevskii. A general way to confined stationary Vlasov-Poisson plasma configurations. Kinetic & Related Models, 2021, 14 (2) : 257-282. doi: 10.3934/krm.2021004
References:
[1]

K. Akô, On the Dirichlet problem for quasi-linear elliptic differential equations of the second order, J. Math. Soc. Japan, 13 (1961), 45-62.  doi: 10.2969/jmsj/01310045.  Google Scholar

[2]

A. A. Arsen'ev, Existence in the large of a weak solution of Vlasov's system of equations, U.S.S.R. Comput. Math. Math. Phys., 15 (1975), 131-143.   Google Scholar

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C. Bardos and P. Degond, Global existence for the Vlasov-Poisson equation in $3$ space variables with small initial data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 101-118.  doi: 10.1016/S0294-1449(16)30405-X.  Google Scholar

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J. BattW. Faltenbacher and E. Horst, Stationary spherically symmetric models in stellar dynamics, Arch. Rat. Mech. Anal., 93 (1986), 159-183.  doi: 10.1007/BF00279958.  Google Scholar

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J. BattE. Jörn and Y. Li, Stationary solutions of the flat Vlasov-Poisson system, Arch. Rat. Mech. Anal., 231 (2019), 189-232.  doi: 10.1007/s00205-018-1277-6.  Google Scholar

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Y. O. Belyaeva, Stationary solutions of the Vlasov-Poisson system for two-component plasma under an external magnetic field in a half-space, Math. Model. Nat. Phenom., 12 (2017), 37-50.  doi: 10.1051/mmnp/2017073.  Google Scholar

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Y. O. Belyaeva and A. L. Skubachevskii, Unique solvability of the first mixed problem for the Vlasov–Poisson system in infinite cylinder, J.Mathem. Sciences, 244 (2020), 930-945.   Google Scholar

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W. Braun and K. Hepp, The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles, Comm. Math. Phys., 56 (1977), 101-113.  doi: 10.1007/BF01611497.  Google Scholar

[10]

S. CaprinoG. Cavallaro and C. Marchioro, Remark on a magnetically confined plasma with infinite charge, Rend. di Matem. Ser. VII., 35 (2014), 69-98.   Google Scholar

[11]

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

[12]

R. de la Llave and R. Obaya, Regularity of the composition operator in spaces of Hölder functions, Discrete Contin. Dynam. Systems, 5 (1999), 157-184.  doi: 10.3934/dcds.1999.5.157.  Google Scholar

[13]

R. J. DiPerna and P. L. Lions, Solutions globales d'equations du type Vlasov–Poisson, C. R. Acad. Sci. Paris Ser. I Math., 307 (1988), 655-658.   Google Scholar

[14]

R. L. Dobrushin, Vlasov equations, Funct. Anal. Appl., 13 (1979), 48-58.   Google Scholar

[15]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Berlin, New-York, Springer, 1977.  Google Scholar

[16]

C. Greengard and P.-A. Raviart, A boundary value problem for the stationary Vlasov-Poisson equations: The plane diode, Comm. Pure Appl. Math., 43 (1990), 473-507.  doi: 10.1002/cpa.3160430404.  Google Scholar

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R. D. Hazeltine and J. D. Meiss, Plasma Confinement, Courier Corporation, 2003. Google Scholar

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E. Horst and R. Hunze, Weak solutions of the initial value problem for the unmodified non-linear Vlasov equation, Math. Methods Appl. Sci., 6 (1984), 262-279.  doi: 10.1002/mma.1670060118.  Google Scholar

[20]

E. Horst, On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation I. General theory, Math. Methods Appl. Sci., 3 (1981), 229-248.  doi: 10.1002/mma.1670030117.  Google Scholar

[21]

H. J. Hwang and J. J. L. Velázquez, On global existence for the Vlasov-Poisson system in a half space, J. Diff. Equ., 247 (2009), 1915-1948.  doi: 10.1016/j.jde.2009.06.004.  Google Scholar

[22]

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

[23]

P. Knopf, Optimal control of a Vlasov-Poisson plasma by an external magnetic field, Calc. Var. Partial Differential Equations, 57 (2018), 134-171.  doi: 10.1007/s00526-018-1407-x.  Google Scholar

[24]

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

[25]

V. P. Maslov, Equations of the self-consistent field, Current problems in mathematics, 11 (1978), 153-234.   Google Scholar

[26]

K. R. Meyer, G. R. Hall and D. Offin, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, Applied Mathematical Sciences 90, Springer New York, 2009.  Google Scholar

[27]

K. Miyamoto, Fundamentals of Plasma Physics and Controlled Fussion, Iwanami Book Service Centre, Tokio, 1997. Google Scholar

[28]

M. Montenegro and A. C. Ponce, The sub-supersolution method for weak solutions, Proc. Amer. Math. Soc., 136 (2008), 2429-2438.  doi: 10.1090/S0002-9939-08-09231-9.  Google Scholar

[29]

K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Diff. Equ., 95 (1992), 281-303.  doi: 10.1016/0022-0396(92)90033-J.  Google Scholar

[30]

S. I. Pokhozhaev, On stationary solutions of the Vlasov-Poisson equations, Differ. Equ., 46 (2010), 530-537.  doi: 10.1134/S0012266110040087.  Google Scholar

[31]

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

[32]

J. Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Comm. Partial Differential Equations, 16 (1991), 1313-1335.  doi: 10.1080/03605309108820801.  Google Scholar

[33]

A. L. Skubachevskii, On the unique solvability of mixed problems for the system of Vlasov-Poisson equations in a half-space, Dokl. Math., 85 (2012), 255-258.  doi: 10.1134/S1064562412020263.  Google Scholar

[34]

A. L. Skubachevskii, Initial–Boundary Value Problems for the Vlasov-Poisson equations in a half-space, Proc. Steklov Inst. Math., 283 (2013), 197-225.  doi: 10.1134/S0081543813080142.  Google Scholar

[35]

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

[36]

A. L. Skubachevskii and Y. Tsuzuki, Classical solutions of the Vlasov-Poisson equations with external magnetic field in a half-space, Comput. Math. Math. Phys., 57 (2017), 541-557.  doi: 10.1134/S0965542517030137.  Google Scholar

[37]

W. M. Stacey, Fusion Plasma Physics, Physics textbook Wiley-VCH, 2nd edition, 2012. Google Scholar

[38]

V. V. Vedenyapin, Boundary value problems for a stationary Vlasov equation, Dokl. Akad. Nauk SSSR, 290 (1986), 777-780.   Google Scholar

[39]

V. V. Vedenyapin, Classification of stationary solutions of the Vlasov equation on a torus and a boundary value problem, Russ. Acad. Sci. Dokl. Math., 45 (1993), 459-462.   Google Scholar

[40]

A. A. Vlasov, Vibrational properties of the electronic gas, Zh. Eksper. Teoret. Fiz., 8 (1938), 291-318.   Google Scholar

[41]

J. Weber, Confined steady states of the relativistic Vlasov–Maxwell system in an infinitely long cylinder, Kinet. Relat. Models, 13(6) (2020), 1135-1161.  doi: 10.3934/krm.2020040.  Google Scholar

[42]

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

show all references

References:
[1]

K. Akô, On the Dirichlet problem for quasi-linear elliptic differential equations of the second order, J. Math. Soc. Japan, 13 (1961), 45-62.  doi: 10.2969/jmsj/01310045.  Google Scholar

[2]

A. A. Arsen'ev, Existence in the large of a weak solution of Vlasov's system of equations, U.S.S.R. Comput. Math. Math. Phys., 15 (1975), 131-143.   Google Scholar

[3]

C. Bardos and P. Degond, Global existence for the Vlasov-Poisson equation in $3$ space variables with small initial data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 101-118.  doi: 10.1016/S0294-1449(16)30405-X.  Google Scholar

[4]

J. Batt, Global symmetric solutions of the initial value problem of stellar dynamics, J. Diff. Equ., 25 (1977), 342-364.  doi: 10.1016/0022-0396(77)90049-3.  Google Scholar

[5]

J. BattW. Faltenbacher and E. Horst, Stationary spherically symmetric models in stellar dynamics, Arch. Rat. Mech. Anal., 93 (1986), 159-183.  doi: 10.1007/BF00279958.  Google Scholar

[6]

J. BattE. Jörn and Y. Li, Stationary solutions of the flat Vlasov-Poisson system, Arch. Rat. Mech. Anal., 231 (2019), 189-232.  doi: 10.1007/s00205-018-1277-6.  Google Scholar

[7]

Y. O. Belyaeva, Stationary solutions of the Vlasov-Poisson system for two-component plasma under an external magnetic field in a half-space, Math. Model. Nat. Phenom., 12 (2017), 37-50.  doi: 10.1051/mmnp/2017073.  Google Scholar

[8]

Y. O. Belyaeva and A. L. Skubachevskii, Unique solvability of the first mixed problem for the Vlasov–Poisson system in infinite cylinder, J.Mathem. Sciences, 244 (2020), 930-945.   Google Scholar

[9]

W. Braun and K. Hepp, The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles, Comm. Math. Phys., 56 (1977), 101-113.  doi: 10.1007/BF01611497.  Google Scholar

[10]

S. CaprinoG. Cavallaro and C. Marchioro, Remark on a magnetically confined plasma with infinite charge, Rend. di Matem. Ser. VII., 35 (2014), 69-98.   Google Scholar

[11]

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

[12]

R. de la Llave and R. Obaya, Regularity of the composition operator in spaces of Hölder functions, Discrete Contin. Dynam. Systems, 5 (1999), 157-184.  doi: 10.3934/dcds.1999.5.157.  Google Scholar

[13]

R. J. DiPerna and P. L. Lions, Solutions globales d'equations du type Vlasov–Poisson, C. R. Acad. Sci. Paris Ser. I Math., 307 (1988), 655-658.   Google Scholar

[14]

R. L. Dobrushin, Vlasov equations, Funct. Anal. Appl., 13 (1979), 48-58.   Google Scholar

[15]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Berlin, New-York, Springer, 1977.  Google Scholar

[16]

C. Greengard and P.-A. Raviart, A boundary value problem for the stationary Vlasov-Poisson equations: The plane diode, Comm. Pure Appl. Math., 43 (1990), 473-507.  doi: 10.1002/cpa.3160430404.  Google Scholar

[17]

Y. Guo, Regularity for the Vlasov equations in a half space, Indiana Univ. Math. J., 43 (1994), 255-320.  doi: 10.1512/iumj.1994.43.43013.  Google Scholar

[18]

R. D. Hazeltine and J. D. Meiss, Plasma Confinement, Courier Corporation, 2003. Google Scholar

[19]

E. Horst and R. Hunze, Weak solutions of the initial value problem for the unmodified non-linear Vlasov equation, Math. Methods Appl. Sci., 6 (1984), 262-279.  doi: 10.1002/mma.1670060118.  Google Scholar

[20]

E. Horst, On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation I. General theory, Math. Methods Appl. Sci., 3 (1981), 229-248.  doi: 10.1002/mma.1670030117.  Google Scholar

[21]

H. J. Hwang and J. J. L. Velázquez, On global existence for the Vlasov-Poisson system in a half space, J. Diff. Equ., 247 (2009), 1915-1948.  doi: 10.1016/j.jde.2009.06.004.  Google Scholar

[22]

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

[23]

P. Knopf, Optimal control of a Vlasov-Poisson plasma by an external magnetic field, Calc. Var. Partial Differential Equations, 57 (2018), 134-171.  doi: 10.1007/s00526-018-1407-x.  Google Scholar

[24]

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

[25]

V. P. Maslov, Equations of the self-consistent field, Current problems in mathematics, 11 (1978), 153-234.   Google Scholar

[26]

K. R. Meyer, G. R. Hall and D. Offin, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, Applied Mathematical Sciences 90, Springer New York, 2009.  Google Scholar

[27]

K. Miyamoto, Fundamentals of Plasma Physics and Controlled Fussion, Iwanami Book Service Centre, Tokio, 1997. Google Scholar

[28]

M. Montenegro and A. C. Ponce, The sub-supersolution method for weak solutions, Proc. Amer. Math. Soc., 136 (2008), 2429-2438.  doi: 10.1090/S0002-9939-08-09231-9.  Google Scholar

[29]

K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Diff. Equ., 95 (1992), 281-303.  doi: 10.1016/0022-0396(92)90033-J.  Google Scholar

[30]

S. I. Pokhozhaev, On stationary solutions of the Vlasov-Poisson equations, Differ. Equ., 46 (2010), 530-537.  doi: 10.1134/S0012266110040087.  Google Scholar

[31]

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

[32]

J. Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Comm. Partial Differential Equations, 16 (1991), 1313-1335.  doi: 10.1080/03605309108820801.  Google Scholar

[33]

A. L. Skubachevskii, On the unique solvability of mixed problems for the system of Vlasov-Poisson equations in a half-space, Dokl. Math., 85 (2012), 255-258.  doi: 10.1134/S1064562412020263.  Google Scholar

[34]

A. L. Skubachevskii, Initial–Boundary Value Problems for the Vlasov-Poisson equations in a half-space, Proc. Steklov Inst. Math., 283 (2013), 197-225.  doi: 10.1134/S0081543813080142.  Google Scholar

[35]

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

[36]

A. L. Skubachevskii and Y. Tsuzuki, Classical solutions of the Vlasov-Poisson equations with external magnetic field in a half-space, Comput. Math. Math. Phys., 57 (2017), 541-557.  doi: 10.1134/S0965542517030137.  Google Scholar

[37]

W. M. Stacey, Fusion Plasma Physics, Physics textbook Wiley-VCH, 2nd edition, 2012. Google Scholar

[38]

V. V. Vedenyapin, Boundary value problems for a stationary Vlasov equation, Dokl. Akad. Nauk SSSR, 290 (1986), 777-780.   Google Scholar

[39]

V. V. Vedenyapin, Classification of stationary solutions of the Vlasov equation on a torus and a boundary value problem, Russ. Acad. Sci. Dokl. Math., 45 (1993), 459-462.   Google Scholar

[40]

A. A. Vlasov, Vibrational properties of the electronic gas, Zh. Eksper. Teoret. Fiz., 8 (1938), 291-318.   Google Scholar

[41]

J. Weber, Confined steady states of the relativistic Vlasov–Maxwell system in an infinitely long cylinder, Kinet. Relat. Models, 13(6) (2020), 1135-1161.  doi: 10.3934/krm.2020040.  Google Scholar

[42]

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

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