ENERGY SOLUTION TO A SCHRÖDINGER-POISSON SYSTEM IN THE TWO-DIMENSIONAL WHOLE SPACE

We consider the Cauchy problem of the two-dimensional Schrödinger-Poisson system in the energy class. Though the Newtonian potential diverges at spatial infinity in the logarithmic order, global well-posedness is proven in both defocusing and focusing cases. The key is a decomposition of the nonline...

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Bibliographic Details
Published inSIAM journal on mathematical analysis Vol. 43; no. 5-6; pp. 2719 - 2731
Main Author MASAKI, Satoshi
Format Journal Article
LanguageEnglish
Published Philadelphia, PA Society for Industrial and Applied Mathematics 10.11.2011
Subjects
Energy
Exact sciences and technology
Mathematical analysis
Mathematics
Partial differential equations
Sciences and techniques of general use
Two dimensional space
global well-posedness
Decomposition method
35Q55
Well posed problem
Schrödinger―Newton system
35Q40
nonlinear Schrödinger equation
Schrödinger equation
Cauchy problem
Non linear equation
Perturbation method
One-dimensional calculations
Energy
Mathematical analysis
Nonlinearity
Two-dimensional calculations
Schrödinger―Poisson system
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Summary:We consider the Cauchy problem of the two-dimensional Schrödinger-Poisson system in the energy class. Though the Newtonian potential diverges at spatial infinity in the logarithmic order, global well-posedness is proven in both defocusing and focusing cases. The key is a decomposition of the nonlinearity into a sum of the linear logarithmic potential and a good remainder, which enables us to apply the perturbation method. Our argument can be adapted to the one-dimensional problem. [PUBLICATION ABSTRACT]
ISSN:0036-1410
1095-7154
DOI:10.1137/100792019