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

DOI10.1007/S00605-017-1037-YzbMATH Open1390.35407arXiv1509.04161OpenAlexW2962997915MaRDI QIDQ681476

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Publication date: 12 February 2018

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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 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.

Full work available at URL: https://arxiv.org/abs/1509.04161

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