doi: 10.3934/krm.2022020
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

On solutions of Vlasov-Poisson-Landau equations for slowly varying in space initial data

KIAM, Miusskaya Pl., 4, Moscow 125047, RF

*Corresponding author: Irina Potapenko

Received  January 2022 Revised  May 2022 Early access May 2022

The paper is devoted to analytical and numerical study of solutions to the Vlasov-Poisson-Landau kinetic equations (VPLE) for distribution functions with typical length $ L $ such that $ \varepsilon = r_D/L << 1 $, where $ r_D $ stands for the Debye radius. It is also assumed that the Knudsen number $ \mathrm{K\!n} = l/L = O(1) $, where $ l $ denotes the mean free pass of electrons. We use the standard model of plasma of electrons with a spatially homogeneous neutralizing background of infinitely heavy ions. The initial data is always assumed to be close to neutral. We study an asymptotic behavior of the system for small $ \varepsilon > 0 $. It is known that the formal limit of VPLE at $ \varepsilon = 0 $ does not describe a rapidly oscillating part of the electrical field. Our aim is to fill this gap and to study the behavior of the "true" electrical field near this limit. We show that, in the problem with standard isotropic in velocities Maxwellian initial conditions, there is almost no damping of these oscillations in the collisionless case. An approximate formula for the electrical field is derived and then confirmed numerically by using a simplified BGK-type model of VPLE. Another class of initial conditions that leads to strong oscillations having the amplitude of order $ O(1/\varepsilon ) $ is considered. A formal asymptotic expansion of solution in powers of $ \varepsilon $ is constructed. Numerical solutions of that class are studied for different values of parameters $ \varepsilon $ and $ \mathrm{K\!n} $.

Citation: Alexander Bobylev, Irina Potapenko. On solutions of Vlasov-Poisson-Landau equations for slowly varying in space initial data. Kinetic and Related Models, doi: 10.3934/krm.2022020
References:
[1]

C. Bardos and N. Besse, The Cauchy problem for the Vlasov-Dirac-Benney equation and related issues in fluid mechanics and semi classical limits, Kin. Relat. Models, 6 (2013), 893-917.  doi: 10.3934/krm.2013.6.893.

[2]

C. Bardos and N. Besse, Hamiltonian structure, fluid representation and stability for the Vlasov-Dirac-Benney equation, Hamiltonian Partial Differential Equations and Applications, 75 (2015), 1-30.  doi: 10.1007/978-1-4939-2950-4_1.

[3]

C. Bardos and A. Nouri, Vlasov equation with Dirac potential used in fusion plasma, J. Math. Phys., 53 (2012), 115621, 16 pp. doi: 10.1063/1.4765338.

[4]

O. V. BatishchevV. Y. BychenkovF. DeteringW. RozmusR. SydoraC. E. Capjack and V. N. Novikov, Heat transport and electron distribution function in laser produced with hot spots, Physics of Plasmas, 9 (2002), 2302.  doi: 10.1063/1.1461385.

[5]

A. V. BobylevM. BisiM. GroppiG. Spiga and I. F. Potapenko, A general consistent BGK model for gas mixtures, Kinetic and Related Models, 11 (2018), 1377-1393.  doi: 10.3934/krm.2018054.

[6]

A. BobylevA. BrantovV. BychenkovS. Karpov and I. Potapenko, DSMC modeling of a single hot spot evolution using the Landau-Fokker-Planck equation, Acta Appl. Math., 132 (2014), 107-116.  doi: 10.1007/s10440-014-9940-x.

[7]

A. V. Bobylev and I. F. Potapenko, Long wave asymptotics for Vlasov-Poisson-Landau kinetic equation, J. Stat. Phys., 175 (2019), 1-18.  doi: 10.1007/s10955-019-02253-z.

[8]

A. V. BrantovV. Yu. BychenkovO. V. Batishchev and W. Rozmus, Nonlocal heat wave propagation due to skin layer plasma heating by short laser pulses, Computer Physics communications, 164 (2004), 67-72.  doi: 10.1016/j.cpc.2004.06.009.

[9]

V. Y. BychenkovW. RozmusV. T. Tikhonchuk and A. V. Brantov, Nonlocal electron transport in a plasma, Phys. Rev. Lett., 75 (1995), 4405-4408.  doi: 10.1103/PhysRevLett.75.4405.

[10]

P. Degond and F. Deluzet, Asymptotic-preserving methods and multiscale models for plasma physics, J. Comput. Phys., 336 (2017), 429-457.  doi: 10.1016/j.jcp.2017.02.009.

[11]

E. M. Epperlein and R. W. Short, A practical nonlocal model for electron heat transport in laser plasmas, Phys. Fluids B, 3 (1991), 3092-3098.  doi: 10.1063/1.859789.

[12]

E. Grenier, Oscillations in quasi-neutral plasma, Commun. in PDEs, 21 (1996), 363-394.  doi: 10.1080/03605309608821189.

[13]

S. GuissetS. BrullB. DubrocaE. d'HumieresS. Karpov and I. Potapenko, Asymptotic-preserving scheme for the M1-Maxwell system in the quasi-neutral regime, Commun. Comput. Phys., 19 (2016), 301-328.  doi: 10.4208/cicp.131014.030615a.

[14]

D. Han-Kwan and T. T. Nguyen, Ill-posedness of the hydrostatic Euler and singular Vlasov equations, Arch. Ration. Mech. Anal., 221 (2016), 1317-1344.  doi: 10.1007/s00205-016-0985-z.

[15]

D. Han-Kwan and F. Rousset, Quasi neutral limit for Vlasov-Poisson with Penrose stable data, Ann. Sci. École Norm. Sup., 49 (2016), 1445–1495. doi: 10.24033/asens.2313.

[16]

L. D. Landau, Kinetic equation for the case of Coulomb interaction, Phys. Zs. Sov. Union, 10 (1936), 154-164. 

[17]

L. D. Landau, On vibrations of electron plasma, Acad. Sci. USSR. J. Phys., 10 (1946), 25-34. 

[18]

E. M. Lifshitz and L. P. Pitaevskii, Course of Theoretical Physics, Landau-Lifshitz, vol. 10, Pergamon Press, Oxford-Elmstord, NY, 1981

[19]

A. A. Vlasov, On vibrational properties of electron gas (in Russian)., J. Exp. Theor Phys. (JETP), 8 (1938), 721-733. 

show all references

References:
[1]

C. Bardos and N. Besse, The Cauchy problem for the Vlasov-Dirac-Benney equation and related issues in fluid mechanics and semi classical limits, Kin. Relat. Models, 6 (2013), 893-917.  doi: 10.3934/krm.2013.6.893.

[2]

C. Bardos and N. Besse, Hamiltonian structure, fluid representation and stability for the Vlasov-Dirac-Benney equation, Hamiltonian Partial Differential Equations and Applications, 75 (2015), 1-30.  doi: 10.1007/978-1-4939-2950-4_1.

[3]

C. Bardos and A. Nouri, Vlasov equation with Dirac potential used in fusion plasma, J. Math. Phys., 53 (2012), 115621, 16 pp. doi: 10.1063/1.4765338.

[4]

O. V. BatishchevV. Y. BychenkovF. DeteringW. RozmusR. SydoraC. E. Capjack and V. N. Novikov, Heat transport and electron distribution function in laser produced with hot spots, Physics of Plasmas, 9 (2002), 2302.  doi: 10.1063/1.1461385.

[5]

A. V. BobylevM. BisiM. GroppiG. Spiga and I. F. Potapenko, A general consistent BGK model for gas mixtures, Kinetic and Related Models, 11 (2018), 1377-1393.  doi: 10.3934/krm.2018054.

[6]

A. BobylevA. BrantovV. BychenkovS. Karpov and I. Potapenko, DSMC modeling of a single hot spot evolution using the Landau-Fokker-Planck equation, Acta Appl. Math., 132 (2014), 107-116.  doi: 10.1007/s10440-014-9940-x.

[7]

A. V. Bobylev and I. F. Potapenko, Long wave asymptotics for Vlasov-Poisson-Landau kinetic equation, J. Stat. Phys., 175 (2019), 1-18.  doi: 10.1007/s10955-019-02253-z.

[8]

A. V. BrantovV. Yu. BychenkovO. V. Batishchev and W. Rozmus, Nonlocal heat wave propagation due to skin layer plasma heating by short laser pulses, Computer Physics communications, 164 (2004), 67-72.  doi: 10.1016/j.cpc.2004.06.009.

[9]

V. Y. BychenkovW. RozmusV. T. Tikhonchuk and A. V. Brantov, Nonlocal electron transport in a plasma, Phys. Rev. Lett., 75 (1995), 4405-4408.  doi: 10.1103/PhysRevLett.75.4405.

[10]

P. Degond and F. Deluzet, Asymptotic-preserving methods and multiscale models for plasma physics, J. Comput. Phys., 336 (2017), 429-457.  doi: 10.1016/j.jcp.2017.02.009.

[11]

E. M. Epperlein and R. W. Short, A practical nonlocal model for electron heat transport in laser plasmas, Phys. Fluids B, 3 (1991), 3092-3098.  doi: 10.1063/1.859789.

[12]

E. Grenier, Oscillations in quasi-neutral plasma, Commun. in PDEs, 21 (1996), 363-394.  doi: 10.1080/03605309608821189.

[13]

S. GuissetS. BrullB. DubrocaE. d'HumieresS. Karpov and I. Potapenko, Asymptotic-preserving scheme for the M1-Maxwell system in the quasi-neutral regime, Commun. Comput. Phys., 19 (2016), 301-328.  doi: 10.4208/cicp.131014.030615a.

[14]

D. Han-Kwan and T. T. Nguyen, Ill-posedness of the hydrostatic Euler and singular Vlasov equations, Arch. Ration. Mech. Anal., 221 (2016), 1317-1344.  doi: 10.1007/s00205-016-0985-z.

[15]

D. Han-Kwan and F. Rousset, Quasi neutral limit for Vlasov-Poisson with Penrose stable data, Ann. Sci. École Norm. Sup., 49 (2016), 1445–1495. doi: 10.24033/asens.2313.

[16]

L. D. Landau, Kinetic equation for the case of Coulomb interaction, Phys. Zs. Sov. Union, 10 (1936), 154-164. 

[17]

L. D. Landau, On vibrations of electron plasma, Acad. Sci. USSR. J. Phys., 10 (1946), 25-34. 

[18]

E. M. Lifshitz and L. P. Pitaevskii, Course of Theoretical Physics, Landau-Lifshitz, vol. 10, Pergamon Press, Oxford-Elmstord, NY, 1981

[19]

A. A. Vlasov, On vibrational properties of electron gas (in Russian)., J. Exp. Theor Phys. (JETP), 8 (1938), 721-733. 

Figure 1.  Electrical field $ E(1.5 \pi, t) $ as a function of time for different parameters $ 0 \leq \varepsilon \leq 1 $
Figure 2.  $ \Delta E(\pi/2, t; \varepsilon ) $ as a function of time for different values of parameter $ 0 \leq \varepsilon \leq 1 $, $ \mathrm{K\!n} = \infty $
Figure 3.  Electrical field $ E(x, t) $, $ x = \pi $ - light grey line, and $ x = 3 \pi/2 $ - dark grey line, $ \partial p/\partial x $, $ x = 3 \pi/2 $ - black line; $ \mathrm{K\!n} = \infty $, $ \varepsilon = 0.04 $ and $ A = 0.5 $
Figure 4.  Electrical field $ E(x, t) $ - dark grey line, $ - \partial p/\partial x $ - black line are partly depicted at a larger scale; $ x = 3 \pi/2 $, $ \mathrm{K\!n} = \infty $, $ \varepsilon = 0.04 $ and $ A = 0.5 $
Figure 5.  Electrical field dependence on time; $ \mathrm{K\!n} = \infty $ - grey lines; $ \mathrm{K\!n} = 1 $ - black lines. Left: $ x = \pi $. Right: $ E $, $ -\partial p/\partial x $, $ x = \pi/2 $
Figure 6.  Time dependence of $ \partial p/\partial x $ for $ \mathrm{K\!n} = \infty $ and $ \mathrm{K\!n} = 1 $, $ x = \pi/2 $
Figure 7.  $ E(x, t) $ for the initial function $ f^0_1 $ with the initial conditions (1) and $ T^0_2(x) $
Figure 8.  Electrical field $ E(3\pi/2, t) $ for $ \mathrm{K\!n} = 0.1, 0.05, 0.025 $; $ \varepsilon = 10^{-3}, A = 5 $, $ x = 3\pi/2 $ - blue line, $ \pi/2 $ - grey line
[1]

Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 361-380. doi: 10.3934/dcds.2002.8.361

[2]

Qiangchang Ju, Hailiang Li, Yong Li, Song Jiang. Quasi-neutral limit of the two-fluid Euler-Poisson system. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1577-1590. doi: 10.3934/cpaa.2010.9.1577

[3]

Franco Flandoli, Marta Leocata, Cristiano Ricci. The Vlasov-Navier-Stokes equations as a mean field limit. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3741-3753. doi: 10.3934/dcdsb.2018313

[4]

T. Candan, R.S. Dahiya. Oscillation of mixed neutral differential equations with forcing term. Conference Publications, 2003, 2003 (Special) : 167-172. doi: 10.3934/proc.2003.2003.167

[5]

Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic and Related Models, 2021, 14 (2) : 211-255. doi: 10.3934/krm.2021003

[6]

Mihaï Bostan. Asymptotic behavior for the Vlasov-Poisson equations with strong uniform magnetic field and general initial conditions. Kinetic and Related Models, 2020, 13 (3) : 531-548. doi: 10.3934/krm.2020018

[7]

Yemin Chen. Smoothness of classical solutions to the Vlasov-Poisson-Landau system. Kinetic and Related Models, 2008, 1 (3) : 369-386. doi: 10.3934/krm.2008.1.369

[8]

Xianyi Li, Deming Zhu. Comparison theorems of oscillation and nonoscillation for neutral difference equations with continuous arguments. Communications on Pure and Applied Analysis, 2003, 2 (4) : 579-589. doi: 10.3934/cpaa.2003.2.579

[9]

Saroj Panigrahi, Rakhee Basu. Oscillation results for second order nonlinear neutral differential equations with delay. Conference Publications, 2015, 2015 (special) : 906-912. doi: 10.3934/proc.2015.0906

[10]

R.S. Dahiya, A. Zafer. Oscillation theorems of higher order neutral type differential equations. Conference Publications, 1998, 1998 (Special) : 203-219. doi: 10.3934/proc.1998.1998.203

[11]

Osama Moaaz, Omar Bazighifan. Oscillation criteria for second-order quasi-linear neutral functional differential equation. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2465-2473. doi: 10.3934/dcdss.2020136

[12]

Ling Hsiao, Fucai Li, Shu Wang. Combined quasineutral and inviscid limit of the Vlasov-Poisson-Fokker-Planck system. Communications on Pure and Applied Analysis, 2008, 7 (3) : 579-589. doi: 10.3934/cpaa.2008.7.579

[13]

Pierre-Emmanuel Jabin. A review of the mean field limits for Vlasov equations. Kinetic and Related Models, 2014, 7 (4) : 661-711. doi: 10.3934/krm.2014.7.661

[14]

Laurent Boudin, Francesco Salvarani. The quasi-invariant limit for a kinetic model of sociological collective behavior. Kinetic and Related Models, 2009, 2 (3) : 433-449. doi: 10.3934/krm.2009.2.433

[15]

Yuanjie Lei, Linjie Xiong, Huijiang Zhao. One-species Vlasov-Poisson-Landau system near Maxwellians in the whole space. Kinetic and Related Models, 2014, 7 (3) : 551-590. doi: 10.3934/krm.2014.7.551

[16]

Hai-Liang Li, Hongjun Yu, Mingying Zhong. Spectrum structure and optimal decay rate of the relativistic Vlasov-Poisson-Landau system. Kinetic and Related Models, 2017, 10 (4) : 1089-1125. doi: 10.3934/krm.2017043

[17]

Gerasimenko Viktor. Heisenberg picture of quantum kinetic evolution in mean-field limit. Kinetic and Related Models, 2011, 4 (1) : 385-399. doi: 10.3934/krm.2011.4.385

[18]

Immanuel Ben Porat. Local conditional regularity for the Landau equation with Coulomb potential. Kinetic and Related Models, , () : -. doi: 10.3934/krm.2022010

[19]

Anaïs Crestetto, Nicolas Crouseilles, Mohammed Lemou. Kinetic/fluid micro-macro numerical schemes for Vlasov-Poisson-BGK equation using particles. Kinetic and Related Models, 2012, 5 (4) : 787-816. doi: 10.3934/krm.2012.5.787

[20]

Nguyen Minh Man, Nguyen Van Minh. On the existence of quasi periodic and almost periodic solutions of neutral functional differential equations. Communications on Pure and Applied Analysis, 2004, 3 (2) : 291-300. doi: 10.3934/cpaa.2004.3.291

2021 Impact Factor: 1.398

Metrics

  • PDF downloads (115)
  • HTML views (55)
  • Cited by (0)

Other articles
by authors

[Back to Top]