A mean field type flow

TL;DR

This paper studies a gradient flow associated with a mean field type equation, proving its global existence, convergence under certain conditions, and divergence for highly negative initial energy.

Contribution

It introduces a new gradient flow framework for mean field equations and establishes conditions for its long-term behavior and convergence.

Findings

Flow exists for all time

Flow converges to solutions under energy conditions

Flow diverges with highly negative initial energy

Abstract

We consider a gradient flow related to the mean field type equation. First, we show that this flow exists for all time. Next, we prove a compactness result for this flow allowing us to get, under suitable hypothesis on its energy, the convergence of the flow to a solution of the mean field type equation. We also get a divergence result if the energy of the initial data is largely negative.