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

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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, Birkhäuser, 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.

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Acknowledgements

We would like to thank the anonymous referees for their useful suggestions.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Estadual de Campinas, Campinas, SP, CEP 13083-859, Brazil

    Lucas C. F. Ferreira & Julio C. Valencia-Guevara

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  1. Lucas C. F. Ferreira
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  2. Julio C. Valencia-Guevara
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Corresponding author

Correspondence to Lucas C. F. Ferreira.

Additional information

Communicated by A. Constantin.

L. Ferreira was supported by FAPESP and CNPQ, Brazil.

J. Valencia-Guevara was supported by CNPQ, Brazil.

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Ferreira, L.C.F., Valencia-Guevara, J.C. Gradient flows of time-dependent functionals in metric spaces and applications to PDEs. Monatsh Math 185, 231–268 (2018). https://doi.org/10.1007/s00605-017-1037-y

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