This project is intended to the study of recent developments in the theory of optimal transport and its applications to nonlinear nonlocal partial differential equations. These equations arise from systems where there exists conservation of mass and where the individuals of the system maintain an interaction among themselves. In some cases, the model can be identified as a gradient flow of the associated entropy on a infinite dimensional Riemannian manifold with respect to the Wasserstein metric. More specifically, we are interested in the analysis of the existence, uniqueness, well-posedness, asymptotic behaviour and steady-state properties of kinetic models like, the recent fractional versions of the porous medium equation and semidiscrete and discrete versions of the nonlinear Fokker-Plank equation where the solution can be identified as a gradient flow for finite Markov chains). We will analyse how the new non-local metrics defined on the space of probability measures can give us qualitative results for equations of the type $\partial_t\rho + (-\Delta)^s\varphi(\rho)=0$, like the classical Wasserstein metric gives on problems of the type $\partial\rho + \nabla\cdot(\rho\nabla U'(\rho))=0$. Furthermore, discrete versions of these metrics have shown to be very fruitful to generate functional inequalities like log-Sobolev and Talagran for semidiscrete problems as the linear Fokker-Planck case. We will study the non-linear case and see whether is possible to find an associated Lyapunov functinal which allow us to obtain information about the long-time asymptotic behaviour of solutions to the semidiscrete version.
News published in Agência FAPESP Newsletter about the scholarship: