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We consider a one dimensional transport model with nonlocal velocity given bythe Hilbert transform and develop a global well-posedness theory of probabilitymeasure solutions. Both the viscous and non-viscous cases are analyzed. Both inoriginal and in self-similar variables, we express the corresponding equationsas gradient flows with respect to a free energy functional including a singularlogarithmic interaction potential. Existence, uniqueness, self-similarasymptotic behavior and inviscid limit of solutions are obtained in the space$\mathcal{P}_{2}(\mathbb{R})$ of probability measures with finite secondmoments, without any smallness condition. Our results are based on the abstractgradient flow theory developed in \cite{Ambrosio}. An important byproduct ofour results is that there is a unique, up to invariance and translations,global in time self-similar solution with initial data in$\mathcal{P}_{2}(\mathbb{R})$, which was already obtained in\textrm{\cite{Deslippe,Biler-Karch}} by different methods. Moreover, thisself-similar solution attracts all the dynamics in self-similar variables. Thecrucial monotonicity property of the transport between measures in onedimension allows to show that the singular logarithmic potential energy isdisplacement convex. We also extend the results to gradient flow equations withnegative power-law locally integrable interaction potentials.

A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity