Existence of global weak solutions to one‐component vlasov‐poisson and fokker‐planck‐poisson systems in one space dimension with measures as initial data

Yuxi Zheng, Andrew Majda

Research output: Contribution to journalArticle

Abstract

We consider Cauchy problems for the 1‐D one component Vlasov‐Poisson and Fokker‐Planck‐Poisson equations with the initial electron density being in the natural space of arbitrary non‐negative finite measures. In particular, the initial density can be a Dirac measure concentrated on a curve, which we refer to as “electron sheet” initial data. These problems resemble both structurally and functional analytically Cauchy problems for the 2‐D Euler and Navier‐Stokes equations (in vorticity formulation) with vortex sheet initial data. Here, we need to define weak solutions more specifically than usual since the product of a finite measure with a function of bounded variation is involved. We give a natural definition of the product, establish its weak stability, and existence of weak solutions follows. Our concept of weak solutions through the newly defined product is justified since solutions to the Fokker‐Planck‐Poisson equation, the analogue of Navier‐Stokes equation, are shown to converge to weak solutions of the Vlasov‐Poisson equation as the Fokker‐Planck term vanishes. The main difficulty is the aforementioned weak stability which we establish through a careful analysis of the explicit structure of these equations. This is needed because the problem studied here is beyond the range of applicability of the “velocity averaging” compactness methods of DiPerna‐Lions. © 1994 John Wiley & Sons, Inc.

Original languageEnglish (US)
Pages (from-to)1365-1401
Number of pages37
JournalCommunications on Pure and Applied Mathematics
Volume47
Issue number10
DOIs
StatePublished - 1994

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Global Weak Solutions
Weak Solution
Cauchy Problem
Navier-Stokes Equations
Electron
Vortex Sheet
Functions of Bounded Variation
Existence of Weak Solutions
Vorticity
Euler Equations
Paul Adrien Maurice Dirac
Carrier concentration
Averaging
Compactness
Vanish
Vortex flow
Non-negative
Analogue
Converge
Curve

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

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title = "Existence of global weak solutions to one‐component vlasov‐poisson and fokker‐planck‐poisson systems in one space dimension with measures as initial data",
abstract = "We consider Cauchy problems for the 1‐D one component Vlasov‐Poisson and Fokker‐Planck‐Poisson equations with the initial electron density being in the natural space of arbitrary non‐negative finite measures. In particular, the initial density can be a Dirac measure concentrated on a curve, which we refer to as “electron sheet” initial data. These problems resemble both structurally and functional analytically Cauchy problems for the 2‐D Euler and Navier‐Stokes equations (in vorticity formulation) with vortex sheet initial data. Here, we need to define weak solutions more specifically than usual since the product of a finite measure with a function of bounded variation is involved. We give a natural definition of the product, establish its weak stability, and existence of weak solutions follows. Our concept of weak solutions through the newly defined product is justified since solutions to the Fokker‐Planck‐Poisson equation, the analogue of Navier‐Stokes equation, are shown to converge to weak solutions of the Vlasov‐Poisson equation as the Fokker‐Planck term vanishes. The main difficulty is the aforementioned weak stability which we establish through a careful analysis of the explicit structure of these equations. This is needed because the problem studied here is beyond the range of applicability of the “velocity averaging” compactness methods of DiPerna‐Lions. {\circledC} 1994 John Wiley & Sons, Inc.",
author = "Yuxi Zheng and Andrew Majda",
year = "1994",
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journal = "Communications on Pure and Applied Mathematics",
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TY - JOUR

T1 - Existence of global weak solutions to one‐component vlasov‐poisson and fokker‐planck‐poisson systems in one space dimension with measures as initial data

AU - Zheng, Yuxi

AU - Majda, Andrew

PY - 1994

Y1 - 1994

N2 - We consider Cauchy problems for the 1‐D one component Vlasov‐Poisson and Fokker‐Planck‐Poisson equations with the initial electron density being in the natural space of arbitrary non‐negative finite measures. In particular, the initial density can be a Dirac measure concentrated on a curve, which we refer to as “electron sheet” initial data. These problems resemble both structurally and functional analytically Cauchy problems for the 2‐D Euler and Navier‐Stokes equations (in vorticity formulation) with vortex sheet initial data. Here, we need to define weak solutions more specifically than usual since the product of a finite measure with a function of bounded variation is involved. We give a natural definition of the product, establish its weak stability, and existence of weak solutions follows. Our concept of weak solutions through the newly defined product is justified since solutions to the Fokker‐Planck‐Poisson equation, the analogue of Navier‐Stokes equation, are shown to converge to weak solutions of the Vlasov‐Poisson equation as the Fokker‐Planck term vanishes. The main difficulty is the aforementioned weak stability which we establish through a careful analysis of the explicit structure of these equations. This is needed because the problem studied here is beyond the range of applicability of the “velocity averaging” compactness methods of DiPerna‐Lions. © 1994 John Wiley & Sons, Inc.

AB - We consider Cauchy problems for the 1‐D one component Vlasov‐Poisson and Fokker‐Planck‐Poisson equations with the initial electron density being in the natural space of arbitrary non‐negative finite measures. In particular, the initial density can be a Dirac measure concentrated on a curve, which we refer to as “electron sheet” initial data. These problems resemble both structurally and functional analytically Cauchy problems for the 2‐D Euler and Navier‐Stokes equations (in vorticity formulation) with vortex sheet initial data. Here, we need to define weak solutions more specifically than usual since the product of a finite measure with a function of bounded variation is involved. We give a natural definition of the product, establish its weak stability, and existence of weak solutions follows. Our concept of weak solutions through the newly defined product is justified since solutions to the Fokker‐Planck‐Poisson equation, the analogue of Navier‐Stokes equation, are shown to converge to weak solutions of the Vlasov‐Poisson equation as the Fokker‐Planck term vanishes. The main difficulty is the aforementioned weak stability which we establish through a careful analysis of the explicit structure of these equations. This is needed because the problem studied here is beyond the range of applicability of the “velocity averaging” compactness methods of DiPerna‐Lions. © 1994 John Wiley & Sons, Inc.

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