From Poincar\'e to logarithmic Sobolev inequalities: a gradient flow   approach
  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. 
Jean Dolbeault (CEREMADE), Bruno Nazaret (CEREMADE), Giuseppe Savar\'e
math.AP