From Poincar\'e to logarithmic Sobolev inequalities: a gradient flow approach

TL;DR

This paper presents a unified gradient flow framework for analyzing drift-diffusion equations, connecting Poincaré and logarithmic Sobolev inequalities through entropy methods.

Contribution

It introduces a novel approach using specific distances to reveal the gradient flow structure of a broad class of entropy-driven drift-diffusion equations.

Findings

Establishes the gradient flow structure for various entropy functionals.

Derives functional inequalities from entropy and entropy production comparisons.

Provides a unified framework for studying the Kolmogorov-Fokker-Planck equation.

Abstract

We use the distances introduced in a previous joint paper to exhibit the gradient flow structure of some drift-diffusion equations for a wide class of entropy functionals. Functional inequalities obtained by the comparison of the entropy with the entropy production functional reflect the contraction properties of the flow. Our approach provides a unified framework for the study of the Kolmogorov-Fokker-Planck (KFP) equation.