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 P 2 (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 timedependent functionals. For that matter, we need to consider some residual terms, timeversions of concepts like λ-convexity, time-differentiability of minimizers for MoreauYosida 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 λ-convexity that changes as the time evolves. AMS MSC2010: 35R20, 34Gxx, 58Exx, 49Q20, 49J40, 35Qxx, 35K15, 60J60, 28A33.

This paper concerns periodic solutions for a 1D-model with nonlocal velocity given by the periodic Hilbert transform. There is a rich literature showing that this model presents singular behavior of solutions via numerics and mathematical approaches. For instance, they can blow up by forming mass-concentration. We develop a global wellposedness theory for periodic measure initial data that allows, in particular, to analyze how the model evolves from those singularities. Our results are based on periodic mass transport theory and the abstract gradient flow theory in metric spaces developed by Ambrosio et al. [2]. A viscous version of the model is also analyzed and inviscid limit properties are obtained.

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We consider a fractional porous medium equation that extends the classical porous medium and fractional heat equations. The flow is studied in the space of periodic probability measures endowed with a non-local transportation distance constructed in the spirit of the Benamou-Brenier formula. For initial periodic probability measures, we show the existence of absolutely continuous curves that are generalized minimizing movements associated to Rényi entropy. For that, we need to obtain entropy and distance properties and to develop a subdifferential calculus in our setting.

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