We consider a one dimensional transport model with nonlocal velocity given by the Hilbert transform and develop a global well-posedness theory of probability measure solutions. Both the viscous and non-viscous cases are analyzed. Both in original and in self-similar variables, we express the corresponding equations as gradient flows with respect to a free energy functional including a singular logarithmic interaction potential. Existence, uniqueness, self-similar asymptotic behavior and inviscid limit of solutions are obtained in the space P2(R) of probability measures with finite second moments, without any smallness condition. Our results are based on the abstract gradient flow theory developed in [2]. An important byproduct of our results is that there is a unique, up to invariance and translations, global in time self-similar solution with initial data in P2(R), which was already obtained in [17,6] by different methods. Moreover, this self-similar solution attracts all the dynamics in self-similar variables. The crucial monotonicity property of the transport between measures in one dimension allows to show that the singular logarithmic potential energy is displacement convex. We also extend the results to gradient flow equations with negative power-law locally integrable interaction potentials.

In this work, we give a complete description of the asymptotic behaviour of quasi-geostrophic equations in the subcritical range γ ∈ ( 1 2 , 1]. We first show that its solutions simplify asymptotically as t → ∞. More precisely, solutions behave as a particular self-similar solution normalized by the mass as t → ∞ and when the initial data belong toOn the other hand, we show that solutions with initial data in L 2 2γ −1 (R 2 ) decay towards zero as t → ∞ in this space. All results are obtained regardless of the size of the initial condition.

This work considers the Keller-Segel system of parabolic-parabolic type in R n for n 2. We prove existence results in a new framework and with initial data inThis initial data class is larger than the previous ones, e.g.

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.

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