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Generalized logarithmic Hardy-Littlewood-Sobolev inequality

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LogHLS.pdf (215.5Kb)
Date
2019
Publisher city
Paris
Dewey
Analyse
Sujet
entropy methods; logarithmic Hardy-Littlewood-Sobolev inequality; drift-diffusion-Poisson equation; nonlinear parabolic equations
Journal issue
International Mathematics Research Notices
Publication date
2019
Article pages
9
Publisher
Oxford University Press
DOI
http://dx.doi.org/10.1093/imrn/rnz324
URI
https://basepub.dauphine.fr/handle/123456789/20108
Collections
  • CEREMADE : Publications
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Author
Dolbeault, Jean
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Li, Xingyu
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Type
Article accepté pour publication ou publié
Abstract (EN)
This paper is devoted to logarithmic Hardy-Littlewood-Sobolev inequalities in the two-dimensional Euclidean space, in presence of an external potential with logarithmic growth. The coupling with the potential introduces a new parameter, with two regimes. The attractive regime reflects the standard logarithmic Hardy-Littlewood-Sobolev inequality. The second regime corresponds to a reverse inequality, with the opposite sign in the convolution term, that allows us to bound the free energy of a drift-diffusion-Poisson system from below. Our method is based on an extension of an entropy method proposed by E. Carlen, J. Carrillo and M. Loss, and on a nonlinear diffusion equation.

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