An invariance principle for gradient flows in the space of probability measures

We seek to establish qualitative convergence results to a general class of evolution PDEs described by gradient flows in optimal transportation distances. These qualitative convergence results come from dynamical systems under the general name of LaSalle Invariance Principle. By combining some of the basic notions of gradient flow theory and dynamical systems, we are able to reproduce this invariance principle in the setting of evolution PDEs under general assumptions. We apply this abstract theory to a non-exhaustive list of examples that recover, simplify, and even extend the results in their respective literatures.