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

TL;DR

This paper extends gradient-flow theory to time-dependent functionals in metric spaces, enabling analysis of PDEs with evolving coefficients and potentials, and providing well-posedness and asymptotic results.

Contribution

It introduces a framework for gradient flows of time-dependent functionals in metric spaces, generalizing previous static theories to include time-varying convexity and residual terms.

Findings

Established global well-posedness of solutions.

Analyzed asymptotic behavior of gradient flows.

Applied results to PDEs with time-dependent coefficients.

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 for a large class of PDEs with time- dependent coefficients like confinement and interaction potentials and diffusion. Our results can be seen as an extension of those in Ambrosio-Gigli-Savar\'e (2005)[2] to the case of time-dependent functionals. 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 satisfy a type of $\lambda$-convexity that changes as the time evolves.