American Institute of Mathematical Sciences

Advanced
December  2010, 3(4): 729-754. doi: 10.3934/krm.2010.3.729

1D Vlasov-Poisson equations with electron sheet initial data

Dongming Wei 1,

1. 

Department of Mathematics, University of Wisconsin, Madison, WI 53706, United States

Received  April 2010 Revised  September 2010 Published  October 2010

We construct global weak solutions for both one-component and two-component Vlasov-Poisson equations in a single space dimension with electron sheet initial data. We give an explicit formula of the weak solution of the one-component Vlasov-Poisson equation provided the electron sheet remains a graph in the $x$-$v$ plane, and we give sharp conditions on whether the moment of this explicit weak solution will blow up or not. We introduce new parameters, which we call "charge indexes", to construct the global weak solution. The moment of the weak solution corresponds to a multi-valued solution to the Euler-Poisson system. Our method guarantees that even if concentration in charge develops, it will disappear immediately. We extend our method to more singular initial data, where charge can concentrate on points at time $t=0$. Examples show that for one-component Vlasov-Poisson equation our weak solution agrees with the continuous fission weak solution, which is the zero diffusion limit of the Fokker-Planck equation. Finally, we propose a novel numerical method to compute solutions of both one-component and two-component Vlasov-Poisson equations and the multi-valued solution of the one-dimensional Euler-Poisson equation.
Keywords: critical thresholds, Vlasov-Poisson equation, Euler-Poisson equation, multi-valued solution..
Mathematics Subject Classification: Primary: 35D30; Secondary: 35Q8.
Citation: Dongming Wei. 1D Vlasov-Poisson equations with electron sheet initial data. Kinetic & Related Models, 2010, 3 (4) : 729-754. doi: 10.3934/krm.2010.3.729
References:
[1]

F. Bouchut, Global weak solution of the Vlasov-Poisson system for small electrons mass, Comm PDEs, 16 (1991), 1337-1365.  Google Scholar

[2]

D. Chae and E. Tadmor, On the finite time blow-up of the Euler-Poisson equations in R2, Commun. Math. Sci., 6 (2008), 785-789.  Google Scholar

[3]

B. Cheng and E. Tadmor, Long time existence of smooth solutions for the rapidly rotating shallow-water and Euler equations, SIAM J. Math. Anal., 39 (2008), 1668-1685.  Google Scholar

[4]

R. S. Dziurzynski, "Patches of Electrons and Electron Sheets for the 1D Vlasov-Poisson Equation," Ph.D. dissertation (supervised by A. Majda), Dept. of Math., University of California, Berkeley, CA, 1987. Google Scholar

[5]

S. Engelberg, H. Liu and E. Tadmor, Critical thresholds in Euler-Poisson equations, Indiana Univ. Math. J., 50 (2001), 109-157.  Google Scholar

[6]

S. Jin and X. T. Li, Multi-phase computations of the semiclassical limit of the Schrödinger equation and related problems: Whitham vs Wigner, Physica D, 182 (2003), 46-85.  Google Scholar

[7]

S. Jin, X. M. Liao and X. Yang, The Vlasov-Poisson equations as the semiclassical Limit of the Schrödinger-Poisson equations: A numerical study, Journal of Hyperbolic Differential Equations, 5 (2008), 569-587.  Google Scholar

[8]

X. T. Li, J. G. Wohlbier, S. Jin and J. H. Booske, An Eulerian method for computing multi-valued solutions of the Euler-Poisson equaqtions and applications to wave breaking in klystrons, Phys. Rev. E., 70 (2004), 016502. Google Scholar

[9]

P. L. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Inventiones Mathematicae, 105 (1991), 415-430.  Google Scholar

[10]

H. Liu and E. Tadmor, Spectral dynamics of the velocity gradient field in restricted flows, Comm. Math. Phys., 228 (2002), 435-466.  Google Scholar

[11]

H. Liu and E. Tadmor, Critical thresholds in 2D restricted Euler-Poisson equations, SIAM J. Appl. Math., 63 (2003), 1889-1910.  Google Scholar

[12]

H. Liu, E. Tadmor and D. Wei, Global regularity of the 4D restricted Euler equations, Physica D, 239 (2010), 1225-1231. Google Scholar

[13]

A. Majda, G. Majda and Y. X. Zheng, Concentrations in the one-dimensional Vlasov-Poisson equation I: Temporal development and non-unique weak solutions in the single component case, Physica D, 74 (1994), 268-300.  Google Scholar

[14]

A. Majda, G. Majda and Y. X. Zheng, Concentrations in the one-dimensiolal Vlasov-Poisson equations. II: Screening and the necessity for measure-valued solutions in the two component case, Physica D, 79 (1994), 41-76.  Google Scholar

[15]

E. Tadmor and D. Wei, On the global regularity of sub-critical Euler-Poisson equations with pressure, J. European Math. Society, 10 (2008), 757-769.  Google Scholar

[16]

Y. X. Zheng and A. Majda, Existence of global weak solutions to one-component Vlasov-Poisson and Fokker-Planck-Poisson systems in one space dimension with messures as initial data, Comm. on Pure and Applied Math., XLVII (1994), 1365-1401.  Google Scholar

show all references

References:
[1]

F. Bouchut, Global weak solution of the Vlasov-Poisson system for small electrons mass, Comm PDEs, 16 (1991), 1337-1365.  Google Scholar

[2]

D. Chae and E. Tadmor, On the finite time blow-up of the Euler-Poisson equations in R2, Commun. Math. Sci., 6 (2008), 785-789.  Google Scholar

[3]

B. Cheng and E. Tadmor, Long time existence of smooth solutions for the rapidly rotating shallow-water and Euler equations, SIAM J. Math. Anal., 39 (2008), 1668-1685.  Google Scholar

[4]

R. S. Dziurzynski, "Patches of Electrons and Electron Sheets for the 1D Vlasov-Poisson Equation," Ph.D. dissertation (supervised by A. Majda), Dept. of Math., University of California, Berkeley, CA, 1987. Google Scholar

[5]

S. Engelberg, H. Liu and E. Tadmor, Critical thresholds in Euler-Poisson equations, Indiana Univ. Math. J., 50 (2001), 109-157.  Google Scholar

[6]

S. Jin and X. T. Li, Multi-phase computations of the semiclassical limit of the Schrödinger equation and related problems: Whitham vs Wigner, Physica D, 182 (2003), 46-85.  Google Scholar

[7]

S. Jin, X. M. Liao and X. Yang, The Vlasov-Poisson equations as the semiclassical Limit of the Schrödinger-Poisson equations: A numerical study, Journal of Hyperbolic Differential Equations, 5 (2008), 569-587.  Google Scholar

[8]

X. T. Li, J. G. Wohlbier, S. Jin and J. H. Booske, An Eulerian method for computing multi-valued solutions of the Euler-Poisson equaqtions and applications to wave breaking in klystrons, Phys. Rev. E., 70 (2004), 016502. Google Scholar

[9]

P. L. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Inventiones Mathematicae, 105 (1991), 415-430.  Google Scholar

[10]

H. Liu and E. Tadmor, Spectral dynamics of the velocity gradient field in restricted flows, Comm. Math. Phys., 228 (2002), 435-466.  Google Scholar

[11]

H. Liu and E. Tadmor, Critical thresholds in 2D restricted Euler-Poisson equations, SIAM J. Appl. Math., 63 (2003), 1889-1910.  Google Scholar

[12]

H. Liu, E. Tadmor and D. Wei, Global regularity of the 4D restricted Euler equations, Physica D, 239 (2010), 1225-1231. Google Scholar

[13]

A. Majda, G. Majda and Y. X. Zheng, Concentrations in the one-dimensional Vlasov-Poisson equation I: Temporal development and non-unique weak solutions in the single component case, Physica D, 74 (1994), 268-300.  Google Scholar

[14]

A. Majda, G. Majda and Y. X. Zheng, Concentrations in the one-dimensiolal Vlasov-Poisson equations. II: Screening and the necessity for measure-valued solutions in the two component case, Physica D, 79 (1994), 41-76.  Google Scholar

[15]

E. Tadmor and D. Wei, On the global regularity of sub-critical Euler-Poisson equations with pressure, J. European Math. Society, 10 (2008), 757-769.  Google Scholar

[16]

Y. X. Zheng and A. Majda, Existence of global weak solutions to one-component Vlasov-Poisson and Fokker-Planck-Poisson systems in one space dimension with messures as initial data, Comm. on Pure and Applied Math., XLVII (1994), 1365-1401.  Google Scholar

[1]

Dongfen Bian, Huimin Liu, Xueke Pu. Modulation approximation for the quantum Euler-Poisson equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (8) : 4375-4405. doi: 10.3934/dcdsb.2020292
[2]

Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Discrete & Continuous Dynamical Systems, 2002, 8 (2) : 361-380. doi: 10.3934/dcds.2002.8.361
[3]

Katherine Zhiyuan Zhang. Focusing solutions of the Vlasov-Poisson system. Kinetic & Related Models, 2019, 12 (6) : 1313-1327. doi: 10.3934/krm.2019051
[4]

A. Alexandrou Himonas, Gerard Misiołek, Feride Tiǧlay. On unique continuation for the modified Euler-Poisson equations. Discrete & Continuous Dynamical Systems, 2007, 19 (3) : 515-529. doi: 10.3934/dcds.2007.19.515
[5]

Yeping Li. Existence and some limit analysis of stationary solutions for a multi-dimensional bipolar Euler-Poisson system. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 345-360. doi: 10.3934/dcdsb.2011.16.345
[6]

Ming Mei, Yong Wang. Stability of stationary waves for full Euler-Poisson system in multi-dimensional space. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1775-1807. doi: 10.3934/cpaa.2012.11.1775
[7]

Zhigang Wu, Weike Wang. Pointwise estimates of solutions for the Euler-Poisson equations with damping in multi-dimensions. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 1101-1117. doi: 10.3934/dcds.2010.26.1101
[8]

Yongcai Geng. Singularity formation for relativistic Euler and Euler-Poisson equations with repulsive force. Communications on Pure & Applied Analysis, 2015, 14 (2) : 549-564. doi: 10.3934/cpaa.2015.14.549
[9]

Blanca Ayuso, José A. Carrillo, Chi-Wang Shu. Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system. Kinetic & Related Models, 2011, 4 (4) : 955-989. doi: 10.3934/krm.2011.4.955
[10]

Silvia Caprino, Guido Cavallaro, Carlo Marchioro. Time evolution of a Vlasov-Poisson plasma with magnetic confinement. Kinetic & Related Models, 2012, 5 (4) : 729-742. doi: 10.3934/krm.2012.5.729
[11]

Jack Schaeffer. Global existence for the Vlasov-Poisson system with steady spatial asymptotic behavior. Kinetic & Related Models, 2012, 5 (1) : 129-153. doi: 10.3934/krm.2012.5.129
[12]

Gang Li, Xianwen Zhang. A Vlasov-Poisson plasma of infinite mass with a point charge. Kinetic & Related Models, 2018, 11 (2) : 303-336. doi: 10.3934/krm.2018015
[13]

Gianluca Crippa, Silvia Ligabue, Chiara Saffirio. Lagrangian solutions to the Vlasov-Poisson system with a point charge. Kinetic & Related Models, 2018, 11 (6) : 1277-1299. doi: 10.3934/krm.2018050
[14]

Zili Chen, Xiuting Li, Xianwen Zhang. The two dimensional Vlasov-Poisson system with steady spatial asymptotics. Kinetic & Related Models, 2017, 10 (4) : 977-1009. doi: 10.3934/krm.2017039
[15]

Meixia Xiao, Xianwen Zhang. On global solutions to the Vlasov-Poisson system with radiation damping. Kinetic & Related Models, 2018, 11 (5) : 1183-1209. doi: 10.3934/krm.2018046
[16]

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
[17]

Jack Schaeffer. On time decay for the spherically symmetric Vlasov-Poisson system. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021021
[18]

Hyung Ju Hwang, Juhi Jang. On the Vlasov-Poisson-Fokker-Planck equation near Maxwellian. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 681-691. doi: 10.3934/dcdsb.2013.18.681
[19]

Laurent Bernis, Laurent Desvillettes. Propagation of singularities for classical solutions of the Vlasov-Poisson-Boltzmann equation. Discrete & Continuous Dynamical Systems, 2009, 24 (1) : 13-33. doi: 10.3934/dcds.2009.24.13
[20]

Zhiming Liu, Zhijian Yang. Global attractor of multi-valued operators with applications to a strongly damped nonlinear wave equation without uniqueness. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 223-240. doi: 10.3934/dcdsb.2019179

2020 Impact Factor: 1.432

Tools

Metrics

  • PDF downloads (62)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences