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

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 app...

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Bibliographic Details
Published in:arXiv.org 2015-09
Main Authors: Ferreira, Lucas C F, Valencia-Guevara, Julio C
Format: Article
Language:English
Subjects:
Asymptotic properties
Convexity
Flow theory
Interpolation
Time dependence
Well posed problems
Online Access:Get full text
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Summary: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-Savaré (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.
ISSN:2331-8422