Periodic solutions for a 1D-model with nonlocal velocity via mass transport

TL;DR

This paper develops a global well-posedness theory for a 1D nonlocal velocity model with periodic solutions, analyzing singularities, viscous effects, and inviscid limits using mass transport and gradient flow theories.

Contribution

It introduces a novel well-posedness framework for the model with measure initial data, incorporating singularities and viscous effects.

Findings

Established global existence and uniqueness for measure initial data.

Analyzed viscous version and inviscid limit behavior.

Provided insights into solution blow-up and mass concentration phenomena.

Abstract

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 well-posedness 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.