Entropy, flows, inequalities – EFI

This proposal lies at the interface between partial differential equations and probability theory. It aims at developping entropy methods and associated notions and techniques, and at applying them to various models in connection with several fields, such as physics (plasmas, Schrödinger, etc) and mathematical biology (notably the Patlak-Keller-Segel system). Special attention will be devoted to their interpretation and approximations in terms of particle systems. Such techniques are nowadays well understood in classical settings, and we intend to extend them to nonlinear and/or degenerate models, and to systems. The long time behaviour is of special interest to us, and for that we shall develop devoted and efficient tools, for instance in terms of functional inequalities. This proposal is in part intended for young researchers. It will focus on obtaining quantitative and optimal results as much as possible, in view of possible numerical applications that would allow interaction with researchers from other fields.