Energy solution to Schr\"odinger-Poisson system in the two-dimensional whole space

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

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Bibliographic Details
Main Author Masaki, Satoshi
Format Journal Article
LanguageEnglish
Published 24.01.2010
Subjects
Mathematics - Analysis of PDEs
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Summary:We consider the Cauchy problem of the two-dimensional Schr\"odinger-Poisson system in the energy class. Though the Newtonian potential diverges at the 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.
DOI:10.48550/arxiv.1001.4308