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

TL;DR

This paper develops a comprehensive theory for a one-dimensional fluid model with nonlocal velocity, establishing existence, uniqueness, and asymptotic behavior of solutions using gradient flow methods in the space of probability measures.

Contribution

It introduces a global well-posedness framework for the model with nonlocal velocity, including viscous and non-viscous cases, and characterizes self-similar solutions and their attractor properties.

Findings

Existence and uniqueness of solutions in probability measure space.

Identification of a unique self-similar solution that attracts all dynamics.

Extension of results to gradient flows with power-law interaction potentials.

Abstract

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 $\mathcal{P}_{2}(\mathbb{R})$ of probability measures with finite second moments, without any smallness condition. Our results are based on the abstract gradient flow theory developed in \cite{Ambrosio}. 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 $\mathcal{P}_{2}(\mathbb{R})$, which was already obtained in \textrm{\cite{Deslippe,Biler-Karch}} 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.