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  April 2021 Early access  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 and 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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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. 

[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.

[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.

[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. 

[14]

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

[15]

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

[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.

[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.

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[25]

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

[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.

[27]

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

[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.

[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.

[30]

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

[31]

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

[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.

[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.

[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.

[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.

[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.

[37]

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

[38]

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

[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. 

[40]

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

[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.

[42]

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

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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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. 

[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.

[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.

[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. 

[14]

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

[15]

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

[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.

[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.

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[25]

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

[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.

[27]

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

[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.

[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.

[30]

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

[31]

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

[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.

[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.

[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.

[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.

[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.

[37]

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

[38]

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

[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. 

[40]

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

[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.

[42]

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

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