Gradient flows express an evolution principle along to the steepest descent direction of a given functional. This principle leads to a particular class of evolution equations, for which uniqueness can be established under mild regularity assumptions, and for which solutions have special properties. This notion, which is classical in Hilbert spaces, has been extended recently to various situations, to general metric spaces, in particular to the space of probability measures endowed with the so-called Wasserstein distance. It made it possible to revisit so classical PDE’s, like the heat equation, which can be interpreted as a gradient flow for the entropy functional.
This study group will be based on some classical papers, together with more recent contributions. The idea is to address the notion of gradient flow under various standpoints, both abstract and linked with real life applications (the list below is not exhaustive) :
- Gradient flows in Hilbert spaces, non-smooth analysis, notion of sub-differential, maximal monotone operators,
- Gradient flows in metric spaces, minimizing motion,
- Gradient flows in the Wasserstein space
- Interpretation of some PDE’s as gradient flows: Heat equation, Fokker-Plank equation, Keller-Segel system, …
- Application to the modeling of crowd motions (both at the microscopic and macroscopic levels)
- Gradient based algorithm for the minimization of convex functionals.