Generalized logarithmic Hardy-Littlewood-Sobolev inequality

TL;DR

This paper extends logarithmic Hardy-Littlewood-Sobolev inequalities in 2D with an external potential, identifying two regimes and applying an entropy method to analyze a drift-diffusion-Poisson system.

Contribution

It introduces a new parameter regime for these inequalities in the presence of a logarithmic potential and applies an entropy-based approach to bound the free energy from below.

Findings

Identified two regimes: attractive and reverse inequalities.

Bound the free energy of a drift-diffusion-Poisson system from below.

Extended entropy methods to new inequality settings.

Abstract

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.