# American Institute of Mathematical Sciences

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

## 1D Vlasov-Poisson equations with electron sheet initial data

 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.
Citation: Dongming Wei. 1D Vlasov-Poisson equations with electron sheet initial data. Kinetic and 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. [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. [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. [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. [5] S. Engelberg, H. Liu and E. Tadmor, Critical thresholds in Euler-Poisson equations, Indiana Univ. Math. J., 50 (2001), 109-157. [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. [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. [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. [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. [10] H. Liu and E. Tadmor, Spectral dynamics of the velocity gradient field in restricted flows, Comm. Math. Phys., 228 (2002), 435-466. [11] H. Liu and E. Tadmor, Critical thresholds in 2D restricted Euler-Poisson equations, SIAM J. Appl. Math., 63 (2003), 1889-1910. [12] H. Liu, E. Tadmor and D. Wei, Global regularity of the 4D restricted Euler equations, Physica D, 239 (2010), 1225-1231. [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. [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. [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. [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.

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##### References:
 [1] F. Bouchut, Global weak solution of the Vlasov-Poisson system for small electrons mass, Comm PDEs, 16 (1991), 1337-1365. [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. [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. [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. [5] S. Engelberg, H. Liu and E. Tadmor, Critical thresholds in Euler-Poisson equations, Indiana Univ. Math. J., 50 (2001), 109-157. [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. [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. [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. [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. [10] H. Liu and E. Tadmor, Spectral dynamics of the velocity gradient field in restricted flows, Comm. Math. Phys., 228 (2002), 435-466. [11] H. Liu and E. Tadmor, Critical thresholds in 2D restricted Euler-Poisson equations, SIAM J. Appl. Math., 63 (2003), 1889-1910. [12] H. Liu, E. Tadmor and D. Wei, Global regularity of the 4D restricted Euler equations, Physica D, 239 (2010), 1225-1231. [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. [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. [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. [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.
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