Dissipative Evolutions and Convergence to Equilibrium – EVOL

Dissipative evolution equations are key tools for the modeling in Physics, Biology or Mathematics applied to Economics. The study of stationary solutions is not sufficient in most of the cases. As soon as complex dynamics enter into the game or if non-linearities are taken into account, one still can assume some general principles (like the least action principle in mechanics), but one has to face many difficult problems in order to study the evolution. Our purpose is to study these questions with tools from probability theory, functional analysis and calculus of variations, in order to predict the rate of convergence to equilibrium in appropriate norms. The point is to use special Lyapunov functionals, entropies and related functionals, which are connected with the structure of the equations, as it is now at least partially understood from optimal transport theory. Technically the participants of this proposal plan to rely on recent results on functional inequalities which have been recently achieved, in particular in the former IFO ANR project. We shall mostly develop three research directions: non linear evolution equations by entropy methods, hypo-coercive and hypo-elliptic problems, stabilization to equilibrium in situations where the equilibrium is only partially known. The participants to the "EVOL" project belong to several areas of applied mathematics, but they are trained to understand (part of) the culture of the other participants, as they clearly demonstrate in previous collaborations. The project intends to go further in the beneficial exchanges between probabilistic and analytic approaches, as they became an obvious necessity to get a better understanding of the models we are studying. Of course such a project wants to be close to the real world. We intend to choose models which are particularly relevant in Physics, Biology... We are also guided by the significance of the results for practical purposes.