Equivalence between duality and gradient flow solutions for one-dimensional aggregation equations

TL;DR

This paper explores the relationship between duality solutions and gradient flow solutions for one-dimensional aggregation equations, establishing a link and developing an existence and uniqueness theory for duality solutions.

Contribution

It connects the geometric gradient flow approach with duality solutions, providing a new framework for analyzing one-dimensional aggregation equations.

Findings

Established the link between duality and gradient flow solutions.

Developed an existence and uniqueness theory for duality solutions.

Restricted analysis to one-dimensional case due to solution limitations.

Abstract

Existence and uniqueness of global in time measure solution for a one dimensional nonlinear aggregation equation is considered. Such a system can be written as a conservation law with a velocity field computed through a selfconsistant interaction potential. Blow up of regular solutions is now well established for such system. In Carrillo et al. (Duke Math J (2011)), a theory of existence and uniqueness based on the geometric approach of gradient flows on Wasserstein space has been developped. We propose in this work to establish the link between this approach and duality solutions. This latter concept of solutions allows in particular to define a flow associated to the velocity field. Then an existence and uniqueness theory for duality solutions is developped in the spirit of James and Vauchelet (NoDEA (2013)). However, since duality solutions are only known in one dimension, we restrict our study to the one dimensional case.