1D Vlasov-Poisson equations with electron sheet initial data

Dongming Wei

doi:10.3934/krm.2010.3.729

Article Contents

2010, Volume 3, Issue 4: 729-754. Doi: 10.3934/krm.2010.3.729

This issue Previous Article Fluid dynamic limit to the Riemann Solutions of Euler equations: I. Superposition of rarefaction waves and contact discontinuity Next Article

1D Vlasov-Poisson equations with electron sheet initial data

Dongming Wei^1,

1.
Department of Mathematics, University of Wisconsin, Madison, WI 53706

Received: March 31, 2010

Revised: August 31, 2010

Published: November 2010

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Abstract

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.

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Mathematics Subject Classification: Primary: 35D30; Secondary: 35Q83.

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References

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