Functional inequalities: nonlinear flows and entropy methods as a tool for obtaining sharp and constructive results

Interpolation inequalities play an essential role in Analysis with fundamental consequences in Mathematical Physics, Nonlinear Partial Differential Equations (PDEs), Markov Processes, etc., and have a wide range of applications in various areas of Science. Research interests have evolved over the 80 years: while mathematicians were originally focussed on abstract properties (like notions of weak solutions and Cauchy problem in PDEs), more qualitative questions (for instance, bifurcation diagrams, multiplicity of the solutions in PDEs and their qualitative behaviour) progressively emerged. Entropy methods for nonlinear PDEs is a typical example: in some cases, the optimal constant in the inequality can be interpreted as the optimal rate of decay of an entropy for an associated evolution equation. Much more can be learned on the way.
This lecture is intended to give an overview of various results on some Gagliardo-Nirenberg-Sobolev and Caffarelli-Kohn-Nirenberg inequalities obtained during the last decade. It will not be a global picture of an active area of research but more a series of snapshots aiming at the illustration of some emerging tools and new directions of research.