Gradient flows of time-dependent functionals in metric spaces and applications to PDEs

Abstract

We develop a gradient-flow theory for time-dependent functionals defined in abstract metric spaces. Global well-posedness and asymptotic behavior of solutions are provided. Conditions on functionals and metric spaces allow to consider the Wasserstein space \({\mathscr {P}}_{2}({\mathbb {R}}^{d})\) and apply the results to a large class of PDEs with time-dependent coefficients like confinement and interaction potentials. For that matter, we need to consider some residual terms, time-versions of concepts like \(\lambda \)-convexity, time-differentiability of minimizers for Moreau–Yosida approximations, and a priori estimates with explicit time-dependence for De Giorgi interpolation. Here, functionals can be unbounded from below and their sublevels need not be compact. In order to obtain strong convergence, a careful analysis is done by using a type of \(\lambda \)-convexity that changes as the time evolves. Our results can be seen as an extension of those in Ambrosio et al. (Gradient flows: in metric spaces and in the space of probability measures, Birkhauser, Basel, 2005) to the case of time-dependent functionals and Rossi et al. (Ann Sc Norm Super Pisa Cl Sci 7(1):97–169, 2008) to functionals with noncompact sublevels.